UNIVERSITY  OF  CALIFORNIA 
AT  LOS  ANGELES 


THE 

PRODUCTION  OF  ELLIPTIC  INTERFERENCES 
IN  RELATION  TO  INTERFEROMETRY 


BY  CARL  BARUS 

Hazard  Professor  of  Physics,  Brown  University 


WASHINGTON,  D.  C. 

Published  by  the  Carnegie  Institution  of  Washington 
1911 


CARNEGIE   INSTITUTION  OF  WASHINGTON 
PUBLICATION  No.  149 


PRESS  OF  J.  B.  LIPPINCOTT  COMPANY 
PHILADELPHIA,  PA. 


Physics 

Library 


PREFACE 


In  connection  with  my  work  on  the  coronas  as  a  means  for  the  study  of 
nucleation,  I  came  across  a  principle  of  interferometry  which  seemed  of  suf- 
ficient importance  to  justify  special  investigation.  This  has  been  under- 
taken in  the  following  chapters,  and  what  appears  to  be  a  new  procedure 
in  interferometry  of  great  promise  and  varied  application  has  been  devel- 
oped. In  case  of  the  coronas  there  is  a  marked  interference  phenomenon 
superposed  on  the  diffractions.  The  present  method  is  therefore  to  consist 
in  a  simplification  or  systematization  of  this  effect,  by  bringing  two  com- 
plete component  diffraction  spectra,  from  the  same  source  of  light,  to  inter- 
fere. This  may  be  done  in  many  ways,  either  directly,  or  with  a  halved 
transmission  or  reflecting  grating,  or  by  using  modifications  of  the  devices 
of  Jamin,  Michelson,  and  others  for  separating  the  components. 

In  the  direct  method,  chapters  II  and  III,  a  mirror  immediately  behind 
the  grating  returns  the  reflected-diffracted  and  diffracted-reflected  rays,  to 
be  superimposed  for  interference,  producing  a  series  of  phenomena  which  are 
eminently  useful,  in  addition  to  their  great  beauty.  In  fact  the  interfer- 
ometer so  constructed  needs  but  ordinary  plate  glass  and  replica  gratings. 
It  gives  equidistant  fringes,  rigorously  straight,  and  their  distance  apart 
and  inclination  are  thus  measurable  by  ocular  micrometry.  The  fringes  are 
duplex  in  character  and  an  adjustment  may  be  made  whereby  ten  small 
fringes  occupy  the  same  space  in  the  field  as  one  large  fringe,  so  that  sudden 
expansions  within  the  limits  of  the  large  fringe  (as  for  instance  in  mag- 
netostriction) are  determinable.  This  has  not  been  feasible  heretofore. 
Length  and  small  angles  (seconds  of  arc)  are  thus  subject  to  micrometric 
measurement.  Finally  the  interferences  are  very  easily  produced  and  are 
strong  with  white  light,  while  the  spectrum  line  may  be  kept  in  the  field  as 
a  stationary  landmark.  The  limiting  sensitiveness  is  half  the  wave-length 
of  light. 

The  theory  of  these  phenomena  has  been  worked  out  in  its  practical  bear- 
ings, advantageous  instrumental  equipment  has  been  discussed,  and  a 
number  of  incidental  applications  to  test  the  apparatus  have  been  made. 
In  much  of  this  work,  including  that  of  the  first  chapter  on  a  modification 
of  Rowland's  spectrometer,  I  had  the  assistance  of  my  son,  Mr.  Maxwell 
Barus,  before  he  entered  into  the  law. 

The  range  of  measurement  of  an  instrument  like  the  above  is  neces- 
sarily limited  to  about  i  cm.  and  the  component  rays  are  not  separated. 
To  increase  the  range  indefinitely  and  to  separate  the  component  rays,  the 
grating  may  replace  the  symmetrically  oblique  transparent  mirror  of 


IV  PREFACE. 

Michelson's  adjustment,  for  instance.  In  this  way  transmitted-reflected- 
diffracted  and  reflected-transmitted-diffracted  spectra,  or  two  correspond- 
ing diffracted  spectra  returned  by  the  opaque  mirrors  M  and  N,  may  be 
brought  to  interfere.  In  both  cases  the  experiments  as  detailed  in  chapters 
IV  and  V  have  been  strikingly  successful.  The  interference  pattern,  how- 
ever, is  now  of  the  ring  type,  extending  throughout  the  whole  spectrum 
from  red  to  violet  with  the  fixed  spectrum  lines  simultaneously  in  view. 
These  rings  closely  resemble  confocal  ellipses,  and  their  centers  have  thesame 
position  in  all  orders  of  spectra;  but  the  major  axes  of  the  ellipses  are  liable 
to  be  vertical  in  the  first  and  horizontal  in  all  the  higher  orders  of  spectra. 

Again  there  is  an  opportunity  for  coarse  and  fine  adjustment,  inasmuch 
as  the  rings  have  the  usual  sensitive  radial  motion,  as  the  virtual  air-space 
increases  or  decreases,  while  the  centers  simultaneously  drift  as  a  whole, 
across  the  fixed  lines  of  the  spectrum,  from  the  red  to  the  violet  end. 

Drift  and  radial  motion  may  be  regulated  in  any  ratio.  The  general 
investigation  shows  that  three  groups,  each  comprising  a  variety  of  inter- 
ferences, are  possible  and  I  have  worked  out  the  practical  side  of  the  theory 
of  the  phenomenon.  Transparent  silvered  surfaces  are  superfluous,  as  the 
ellipses  are  sufficiently  strong  to  need  no  accessory  treatment.  Consider- 
able width  of  spectrum  slit  is  also  admissible.  Finally,  the  ellipses  may  be 
made  of  any  size  and  the  sensitiveness  of  their  lateral  motion  may  be  regu- 
lated to  any  degree  by  aid  of  a  compensator.  In  this  adjustment  the  drift 
may  be  made  even  more  delicate  than  the  radial  motion,  thus  constituting 
a  new  feature  in  interferometry. 

The  interesting  result  follows  from  the  work  that  the  displacement  of 
the  centers  of  ellipses  does  not  correspond  to  the  zero  of  path  difference,  but 
to  an  adjustment  in  which  /*— Xd/z/ciX  (where  n  is  the  index  of  refraction 
and  X  the  wave-length)  is  critical. 

It  is  obvious  that  the  transparent  plate  grating  may  be  replaced  by  a 
reflecting  grating.  Thus  the  grating  may  be  replaced  by  a  plate  of  glass, 
as  in  Michelson's  case,  and  the  function  of  the  two  opaque  mirrors  may  be 
performed  by  two  identical  plane  reflecting  gratings,  each  symmetrically 
set  at  the  diffraction  angle  of  the  spectrum  to  the  incident  ray.  In  this 
case  the  undeviated  reflection  is  thrown  out,  whereas  the  spectra  overlap 
in  the  telescope.  Finally  the  grating  may  itself  be  cut  in  half  by  a  plane 
parallel  to  the  rulings,  whereupon  the  two  overlapping  spectra  will  interfere 
elliptically,  if  by  a  micrometer  one-half  is  slightly  moved,  parallel  to  itself, 
out  of  the  original  common  vertical  plane  of  the  grating. 

Throughout  the  editorial  work  and  in  the  drawings  I  have  profited  by  the 
aptitude  and  tireless  efficiency  of  my  former  student,  Miss  Ada  I.  Burton. 
But  for  her  self-sacrificing  assistance  it  would  have  been  difficult  to  bring 
the  present  work  to  completion. 

CARL  BARUS. 

BROWN  UNIVERSITY,  PROVIDENCE,  RHODE  ISLAND. 


CONTENTS. 


CHAPTER  I. — On  an  Adjustment  of  the  Plane  Grating  Similar  to  Row- 
land's method  for  the  Concave  Grating.     By  C.  BARUS  AND  M.  BARUS. 

Page 

1 .  Apparatus.     Figs,  i  and  2 1-2 

2 .  Single  focussing  lens  in  front  of  grating 2-3 

3.  Adjustments.     Figs.  3  and  4,  A,  B,  C 3-6 

4.  Data  for  single  lens  in  front  of  grating.     Table  i 6 

5.  Single  focussing  lens  behind  grating 6 

6.  Data  for  single  lens  behind  grating.     Table  2 6 

7 .  Collimator  method 7 

8.  Data  for  collimator  method.     Table  3 7 

9.  Discussion 8-9 

10.  Reflecting  plate  grating 9 

11.  Rowland's  concave  grating.     Table  4 10-1 1 

1 2 .  Summary 1 1 

CHAPTER  II.— The  Interference  of  the  Reflected-Diffracted  and  the  Dif- 
fracted-Reflected  Rays  of  a  Plane  Transparent  Grating,  and  on  an 
Interferometer.  By  C.  BARUS  AND  M.  BARUS. 

13.  Introductory.     Figs.  5,  6,  7 I3~I5 

14.  Observations.     Table  5 16—17 

15.  Equations I7~I9 

1 6.  Differential  equations 19—20 

17.  Normal  incidence  or  diffraction,  etc 20 

1 8.  Comparison  of  the  equations  of  total  interference  with  observation 20-21 

19.  Interferometer.     Figs.  8  and  9;  table  6 21-25 

20.  Secondary  interferences 25 

21.  Summary  of  secondary  interferences 26 

22.  Convergent  and  divergent  rays 26—27 

23.  Measurement  of  small  horizontal  angles 27-28 

24.  Summary 28 

CHAPTER  III. — The  Grating  Interferometer.     By  C.  BARUS  AND  M.  BARUS. 

25.  Introductory.     Figs.  10  and  1 1 29 

26.  Apparatus 30 

27.  Adjustments 3 1-32 

28.  Angular  extent  of  the  fringes.     Tables  7  and  8 32~34 

29.  Test  made  by  magneto-striction.     Figs.  12  to  14 ;  table  9 34-38 

30.  Summary 38 

CHAPTER  IV. — The   Use  of  the  Grating  in  Interferometry;  Experi- 
mental Results. 

3 1 .  Introductory.     Fig.  15 39-4 1 

32.  Special  properties.     Fig.  16 41—44 

33.  Elementary  theory.     Fig.  17 44~47 

34.  Compensator.     Table  10 47-48 

CHAPTER  V. — Interferometry  with  the  Aid  of  the  Grating;  Theoretical 
Results. 

PART    I.    INTRODUCTION. 

35.  Remarks  on  the  phenomena.     Figs.  18  and  19 49~5i 

36.  Cause  of  ellipses 5 1-53 

37.  The  three  principal  adjustments  for  interference.     Figs.  20,  21,  22,  23. .    53-55 


VI  CONTENTS. 

PART    II.    DIRECT  CASES  OF  INTERFERENCE.       DIFFRACTION  ANTECEDENT. 

38.  Diffraction  before  reflection 55~56 

39.  Elementary  theory.     Figs.  24  and  25 56-58 

40.  Equations  for  the  present  case.     Fig.  26;  table  1 1 58-61 

4  r .  Interferometer 6  r 

42.  Discrepancy  of  the  table 62 

PART  III.    DIRECT  CASES  OF  INTERFERENCE.      REFLECTION   ANTECEDENT. 

43.  Equations  for  this  case.     Tables  12,  13,  and  14;  figs.  27  and  28 62-66 

44.  Divergence  per  fringe.     Tables  15  and  16;  fig.  29 66-70 

45.  Case  of  dtl/dy,  dX/dy,  etc.     Fig.  30 70-72 

46.  Interferometry  in  terms  of  radial  motion 72 

47.  Interferometry  by  displacement 72 

PART    IV.    INTERFERENCES    IN    GENERAL,    AND    SUMMARY. 

48.  The  individual  interferences.     Figs.  31,  32,  33 73~75 

49.  The  combined  interferences.     Table  17 75~?6 

50.  Special  results 76-77 


CHAPTER  I. 


ON  AN  ADJUSTMENT  FOR  THE  PLANE  GRATING  SIMILAR  TO  ROWLAND'S 
METHOD  FOR  THE   CONCAVE  GRATING.     By  C.  Barus  and  M.  Barus. 

1.  Apparatus. — The  remarkable  refinement  which  has  been  attained 
(notably  by  Mr.  Ives  and  others)  in  the  construction  of  celluloid  replicas 
of  the  plane  grating  makes  it  desirable  to  construct  a  simple  apparatus 
whereby  the  spectrum  may  be  shown  and  the  measurement  of  wave-length 
made,  in  a  way  that  does  justice  to  the  astonishing  performance  of  the  grat- 
ing. We  have  therefore  thought  it  not  superfluous  to  devise  the  following 
inexpensive  contrivance,  in  which  the  wave-length  is  strictly  proportional 
to  the  shift  of  the  carriage  at  the  eyepiece;  which  for  the  case  of  a  good 
2 -meter  scale  divided  into  centimeters  admits  of  a  measurement  of  wave- 
length to  a  few  Angstrom  units  and  with  a  millimeter  scale  should  go  much 
further. 

Observations  are  throughout  made  on  both  sides  of  the  incident  rays  and 
from  the  mean  result  most  of  the  usual  errors  should  be  eliminated  by 
symmetry.  It  is  also  shown  that  the  symmetrical  method  may  be  adapted 
to  the  concave  grating. 

In  fig.  i  A  and  B  are  two  double  slides,  like  a  lathe  bed,  155  cm.  long 
and  ii  cm.  apart,  which  happened  to  be  available  for  optical  purposes,  in 
the  laboratory.  They  were  therefore  used,  although  single  slides  at  right 
angles  to  each  other,  similar  to  Rowland's,  would  have  been  preferable. 
The  carriages  C  and  D,  30  cm.  long,  kept  at  a  fixed  distance  apart  by  the 
rod  ab,  are  in  practice  a  length  of  >^-inch  gas  pipe,  swiveled  at  a  and  6,  169.4 
decimeters  apart,  and  capable  of  sliding  right  and  left  and  to  and  fro, 
normally  to  each  other. 

The  swiveling  joint,  which  functioned  excellently,  is  made  very  simply 
of  ^-inch  gas-pipe  tees  and  nipples,  as  shown  in  fig.  2.  The  lower  nipple 
N  is  screwed  tight  into  the  T,  but  all  but  tight  into  the  carriage  D,  so  that 
the  rod  ab  turns  in  the  screw  N,  kept  oiled.  Similarly  the  nipple  N"  is  either 
screwed  tight  into  the  T  (in  one  method,  revolvable  grating),  or  all  but 
tight  (in  another  method,  stationary  grating),  so  that  the  table  tt,  which 
carries  the  grating  g,  may  be  fixed  while  the  nipple  N"  swivels  in  the  T. 

Any  ordinary  laboratory  clamp  K  and  a  similar  one  on  the  upright  C 
(screwed  into  the  carriage  D}  secures  a  small  rod  k  for  this  purpose.  Again 
a  hole  may  be  drilled  through  the  standards  at  K  and  C  and  provided  with 
set  screws  to  fix  a  horizontal  rod  k  or  check.  The  rod,  k,  should  be  long 
enough  to  similarly  fix  the  standard  on  the  slide  5,  carrying  the  slit,  and  be 
prolonged  further  toward  the  rear  to  carry  the  flame  or  Geissler  tube  appa- 

1 


2  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES  \ 

ratus.  The  table  tt  is  revoluble  on  a  brass  rod  fitting  within  the  gas  pipe, 
which  has  been  slotted  across  so  that  the  conical  nut  M  may  hold  it  firmly. 
The  axis  passes  through  the  middle  of  the  grating,  which  is  fastened  cen- 
trally to  the  table  tt  with  the  usual  tripod  adjustment. 


FIG.  i. — Plan  of  the  apparatus. 

2.  Single  focussing  lens  in  front  of  grating. — I  shall  describe  three 
methods  in  succession,  beginning  with  the  first.  Here  a  large  lens  L,  of 
about  56  cm.  focal  distance  and  about  10  cm.  in  diameter,  is  placed  just  in 
front  of  the  grating,  properly  screened  and  throwing  an  image  of  the  slit 
5  upon  the  cross-hairs  of  the  eyepiece  E,  the  line  of  sight  of  which  is  always 
parallel  to  the  rod  ab,  the  end  b  swiveled  in  the  carriage  C,  as  stated.  (See 
fig.  2.)  An  ordinary  lens  of  5  to  10  cm.  focal  distance,  with  an  appropriate 


IN    RELATION    TO    INTERFEROMETRY.  6 

diaphragm,  is  adequate  and  in  many  ways  preferable  to  stronger  eyepieces. 
The  slit,  S,  carried  on  its  own  slide  and  capable  of  being  clamped  to  C  when 
necessary,  as  stated,  is  additionally  provided  with  a  long  rod  hh  lying  under- 
neath the  carriage,  so  that  the  slit  5  may  be  put  accurately  in  focus  by  the 
observer  at  C.  F  is  a  carriage  for  the  mirror  or  the  flame  or  other  source  of 
light  whose  spectrum  is  to  be  examined;  or  the  source  may  be  adjustable 
on  the  rear  of  the  rod  by  which  D  and  5  are  locked  together. 

Finally,  the  slide  AB  is  provided  with  a  scale  55  and  the  position  of  the 
carriage  C  read  off  by  aid  of  the  vernier  v.  A  good  wooden  scale,  graduated 
in  centimeters,  happened  to  be  available,  the  vernier  reading  to  within  one 
millimeter.  For  more  accurate  work  a  brass  scale  in  millimeters  with  an 
appropriate  vernier  has  since  been  provided. 

Eyepiece  E,  slit  5,  flame  F,  etc.,  may  be  raised  and  lowered  by  the  split 
tube  device  shown  as  at  M  and  M'  in  fig.  2 . 


FIG.   2. — Elevation  of  standards  of  grating  (g)  and  eyepiece  (£). 

3.  Adjustments. — The  first  general  test  which  places  slit,  grating  and  its 
spectra,  and  the  two  positions  of  the  eyepiece  in  one  plane,  is  preferably 
made  with  a  narrow  beam  of  sunlight,  though  lamplight  suffices  in  the  dark. 
Thereafter  let  the  slit  be  focussed  with  the  eyepiece  on  the  right,  marking 
the  position  of  the  slit ;  next  focus  the  slit  for  the  eyepiece  on  the  left ;  then 
place  the  slit  midway  between  these  positions  and  now  focus  by  slowly 


4  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

rotating  the  grating.  The  slit  will  then  be  found  in  focus  for  both  positions 
and  the  grating  which  acts  as  a  concave  lens  counteracting  L  will  be  sym- 
metrical with  respect  to  both  positions.  Let  the  grating  be  thus  adjusted 
when  fixed  normally  to  the  slide  B  or  parallel  to  A .  Then  for  the  first  order 
of  the  spectra  the  wave-length  X  =  d  sin  B,  where  d  is  the  grating  space  and  B 
the  angle  of  diffraction.  The  angle  of  incidence  *  is  zero. 

Again  let  the  grating  adjusted  for  symmetry  be  free  to  rotate  with  the 
rod  ab.  Then  6  is  zero  and  \  =  d  sin  i, 

In  both  cases,  however,  if  2*  be  the  distance  apart  of  the  carriage  C, 
measured  on  the  scale  ss,  for  the  effective  length  of  rod  ab  =  r  between  axis 
and  axis, 

\  =  dx/r  or  (d/2r)zx 

so  that  in  either  case  X  and  x  are  proportional  quantities. 


FIG.  3. — A,  B,  C,  Diagrams  relative  to  conjugate  foci. 


The  whole  spectrum  is  not,  however,  clearly  in  focus  at  one  time,  though 
the  focussing  by  aid  of  the  rod  hh  is  not  difficult.  For  extreme  positions  a 
pulley  adjustment,  operating  on  the  ends  of  h,  is  a  convenience,  the  cords 
running  around  the  slide  A  A.  In  fact  if  the  slit  is  in  focus  when  the  eye- 
piece is  at  the  center  (0  =  0,  *  =  o),  at  a  distance  a  from  the  grating,  then  for 
the  fixed  grating,  fig.  4, 


where  a'  is  the  distance  between  grating  and  slit  for  the  diffraction  corre- 
sponding to  x.  Hence  the  focal  distance  of  the  grating  regarded  as  a  con- 
cave lens  is/'  =  ar2/*.  For  the  fixed  grating  and  a  given  color,  it  frequently 


IN    RELATION    TO    INTERFEROMETRY.  5 

happens  that  the  undeviated  ray  and  the  diffracted  rays  of  the  same  color 
are  simultaneously  in  focus,  though  this  does  not  follow  from  the  equation. 
Again  for  the  rotating  grating,  fig.  3A,  if  a"  is  the  distance  between  slit 

r2  -x2  r2-  x2 

and  grating  a"  =  a    -  §  - ,  so  that  its  focal  distance  is  /"  =  a  — 3 — •    It  follows 

also  that  a'Xa"  =  a.  For  a  =  80  cm.  and  sodium  light,  the  adjustment 
showed  roughly/'  =  650  cm.,  f"  =  570  cm.,  the  behavior  being  that  of  a  weak 
concave  lens.  The  same  a  =  80  cm.  and  sodium  light  showed  furthermore 
a' =  91  and  a" '  =  70.3. 

Finally  there  is  a  correction  needed  for  the  lateral  shift  of  rays,  due  to 
the  fact  that  the  grating  film  is  inclosed  between  two  moderately  thick 


£' 


£" 


FIG.  4. — Diagram  of  adjustment  for  concave  grating.  R,  pa,  and  p' 
are  measured  from  G,  p  and  p0"  from  G'. 


plates  of  glass  (total  thickness  t 
shift  thus  amounts  to 

tx 


.99  cm.)  of  the  index  of  refraction  n.    This 


xi         i  _i \b 

'\vTT^3    vV-sV^/a 


But  since  this  shift  is  on  the  rear  side  of  the  lens  L,  its  effect  on  the  eyepiece 
beyond  will  be  (if  /is  the  principal  focal  distance  and  b  the  conjugate  focal 
distance  between  lens  and  eyepiece,  remembering  that  the  shift  must  be 
resolved  parallel  to  the  scale  ss] 


where  the  correction  e  is  to  be  added  to  ix,  and  is  positive  for  the  rotating 
grating  and  negative  for  the  stationary  grating. 


6 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


Hence  in  the  mean  values  of  2*  for  stationary  and  rotating  grating  the 
effect  of  e  is  eliminated.  For  a  given  lens  at  a  fixed  distance  from  the  eye- 
piece (b/f—  i)  is  constant. 

4.  Data  for  single  lens  in  front  of  grating.— In  conclusion  we  select  a 
few  results  taken  at  random  from  the  notes. 

TABLE  i. 


Stationary  grating. 

Rotating  grating. 

Line. 

Observed 

2X' 

Shift. 

Corrected 
ax 

Line. 

Observed 

3X' 

Shift. 

Corrected 
ax 

c 

132.60 
118.90 
98.23 

87.87 

-  .26 
-•23 
-.19 

-.16 

132.34 
118.67 
98.04 

87.71 

$•:•••':•: 

Hydrogen 
violet.  . 

132.  10 
118.45 
97.90 

87-50 

+  .26 
•  23 
•19 

.16 

132.36 
118.68 
98.09 

87.66 

5E::: 

Hydrogen 
violet.. 

The  real  test  is  to  be  sought  in  the  corresponding  values  of  2X  for  the 
stationary  and  rotating  cases,  and  these  are  very  satisfactory,  remembering 
that  a  centimeter  scale  on  wood  with  a  vernier  reading  to  millimeters  only 
was  used  for  measurement. 

5.  Single  focussing  lens  behind  the  grating.— The  lens  L,  which  should 
be  achromatic,  is  placed  in  the  standard  C.  The  light  which  passes  through 
the  grating  is  now  convergent,  whereas  it  was  divergent  in  §2.  Hence  the 
focal  points  at  distances  a',  a"  lie  in  front  of  the  grating ;  but  in  other  respects 
the  conditions  are  similar  but  reversed.  Apart  from  signs, 

r2  —  £ 
a'  =  a — 5— ,  for  the  stationary  grating 

a"  =  a  -j — 2»  for  the  rotating  grating 

/    •"•  .V 

The  correction  for  shift  loses  the  factor  (&//—  i)  and  becomes 
tx, 


As  intimated,  it  is  negative  for  the  rotating  grating  and  positive  for  the 
stationary  grating.    It  is  eliminated  in  the  mean  values. 

6.  Data  for  single  lens  behind  the  grating. — An  example  of  the  results 
will  suffice.    Different  parts  of  the  spectrum  require  focussing. 


TABLE  2. 

Grating. 

,     Line.             2*' 

Shift.              ax 

Stationary  

D2          118.40 

+  .  13         118.53 

Rotating 

D2           118  65 

—  .  13         118.52 

•*     | 

IN    RELATION    TO    INTERFEROMETRY. 


The  values  of  2x,  remembering  that  a  centimeter  scale  was  used,  are 
again  surprisingly  good.  The  shift  is  computed  by  the  above  equation. 
It  may  be  eliminated  in  the  mean  of  the  two  methods.  The  lens  L'  at  C 
may  be  more  easily  and  firmly  fixed  than  at  L. 

7.  Collimator  method.— The  objection  to  the  above  single-lens  methods 
is  the  fact  that  the  whole  spectrum  is  not  in  sharp  focus  at  once.    Their 
advantage  is  the  simplicity  of  the  means  employed.     If  lenses  at  L'  and 
at  L  are  used  together,  the  former  as  a  collimator  (achromatic)  and  with  a 
focal  distance  of  about  50  cm.,  and  the  latter  (focal  distance  to  be  large,  say 
150  cm.)  as  the  objective  of  a  telescope,  all  the  above  difficulties  disappear 
and  the  magnification  may  be  made  even  excessively  large.    The  whole 
spectrum  is  brilliantly  in  focus  at  once  and  the  corrections  for  the  shift  of 
lines  due  to  the  plates  of  the  grating  vanish.    Both  methods  for  stationary 
and  rotating  gratings  give  identical  results.    The  adjustments  are  easy  and 
certain,  for  with  sunlight  (or  lamplight  in  the  dark)  the  image  of  the  slit 
may  be  reflected  back  from  the  plate  of  the  grating  on  the  plane  of  the  slit 
itself,  while  at  the  same  time  the  transmitted  image  may  be  equally  sharply 
adjusted  on  the  focal  plane  of  the  eyepiece.    It  is  therefore  merely  necessary 
to  place  the  plane  of  spectra  horizontal.     Clearly  a'  and  a"  are  all  infinite. 

In  this  method  the  slides  S  and  D  are  clamped  at  the  focal  distance  apart, 
so  that  flame,  etc.,  slit,  collimator  lens,  and  grating  move  together.  The 
grating  may  or  may  not  be  revoluble  with  the  lens  L  on  the  axis  a. 

8.  Data  for  the  collimator  method.— The  following  data  chosen  at  ran- 
dom may  be  discussed.    The  results  were  obtained  at  different  times  and 
under  different  conditions.    The  grating  nominally  contained  about  15,050 
lines  per  inch.     The  efficient  rod-length  db  was  ^=169.4  cm.     Hence  if 
i/ C=  15, osoX. 3937X338.8,  the  wave-length \—C=2X  cm. 

TABLE  3. — Stationery  rotating  grating. 


Lines. 

2*' 

2X 

D,  

/  "8-30  \ 

\  118.08  / 

IlS.IQ 

Dt  

f  118.27  I 
I  118.05  / 

118.16 

Rowland's  value  of  Dz  is  58.92  X  lo"6  cm. ;  the  mean  of  the  two  values  of 
2x  just  stated  will  give  58.87  X 10"*  cm.  The  difference  may  be  due  either 
to  the  assumed  grating-space,  or  to  the  value  of  R  inserted,  neither  of  which 
was  reliable  absolutely  to  much  within  o.i  per  cent. 

Curiously  enough,  an  apparent  shift  effect  remains  in  the  values  of  zx  for 
stationary  and  rotating  grating,  as  if  the  collimation  were  imperfect.  The 
reason  for  this  is  not  clear,  though  it  must  in  any  case  be  eliminated  in  the 
mean  result.  Possibly  the  friction  involved  in  the  simultaneous  motion 
of  three  slides  is  not  negligible  and  may  leave  the  system  under  slight  strain 
equivalent  to  a  small  lateral  shift  of  the  slit. 


8  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

9.  Discussion. — The  chief  discrepancy  is  the  difference  of  values  for  2* 
in  the  single-lens  system  (for  Dt,  118.7  and  118.5  cm.,  respectively)  as  com- 
pared with  a  double-lens  system  (for  D2,  1 18.2  cm.)  amounting  to  0.2  to  0.4 
per  cent.  For  any  given  method  this  difference  is  consistently  maintained. 
It  does  not  therefore  seem  to  be  mere  chance. 

We  have  for  this  reason  computed  all  the  data  involved  for  a  fixed  grat- 
ing 5  cm.  in  width,  in  the  two  extreme  positions,  fig.  3,  C,  the  ray  beinj^ 
normally  incident  at  the  left-hand  and  the  right-hand  edges,  respectively, 
for  the  method  of  §  6.  The  meaning  of  the  symbols  is  clear  from  fig.  3,  C,  S 
being  the  virtual  source,  g  the  grating,  e  the  diffraction  conjugate  focus  of 
5  for  normal  incidence,  so  that  b  =  r  is  the  fixed  length  of  rod  carrying  grat- 
ing and  eyepiece.  It  is  almost  sufficient  to  assume  that  all  diffracted  rays 
b'  to  b"  are  directed  toward  e, in  which  case  equation  (i)  would  hold;  but 
this  will  not  bring  out  the  divergence  in  question.  They  were  therefore  not 
used.  Hence  the  following  equations  (2)  to  (5)  successively  apply  where 
d  is  the  grating  space. 

cot  6'  =  (big  +  sin  0)/cos  6;     cot  0"  =  (bjg  -  sin  0)/cos  6  ( i ) 

a  =  6/cos20;     a'  =  a"  =  V  g2  +  a2  (2) 

sin  i'  =  sin  i"  =  g/a'  ( 3 ) 

-  sin  i'  +  sin  (6  +  6')  =  Xjd;  sin  6  =  Jl/d;  sin  i"  +  sin  (0  +  6"}  =  l/d  (4) 

cos2*'/a'  =  cos2  (0  +  0')V;     cos2  i  "la"  =  cos2  (6  +  6")  /b"  (5) 

Since  8,  g,  \,  d,  b  are  given,  d'  and  8"  are  found  in  equation  (4),  apart  from 
signs.  If  81  and  5i"  be  the  distance  apart  of  the  projections  of  the  extremi- 
ties of  6'  and  6,  b  and  b",  respectively,  on  the  line  x, 

di  =g+(b  —  b'}  sin  0-6'  sin  i'  d"  =  g  +  (b"  —  6)  sin  0  —  b"  sin  i"     (6) 

If  fo'  and  &*  be  the  distance  apart  of  the  intersections  of  the  prolongation 
of  b'  and  b,  b  and  b",  respectively,  with  the  line  x 

<V  =  sin  (0  +  0')(&cos0/cos  (0  +  0')-&') 
<V  =  sin  (0-0")(&"-&cos0/cos  (6-6"}) 

Given  6=  169.4  cm.,  6  =  20°  22',  about  for  sodium,  #=5  cm.,  the  following 
values  are  obtained : 

0'  =  i°36'  0=192. 7  cm.         0"=i°34'  a'  =a"=  192.8  cm. 

i'=z"  =  i°3o'          6'  =  i66.ocm.  r  =  6=i69.4cm.  6"  =  i72.4cm. 

whence 

#,'  =  i. 92  cm.  o2"=i.74cm. 

These  limits  are  surprisingly  wide.  If,  however,  they  should  be  quite  wiped 
out  on  focussing,  for  any  group  of  rays  and  symmetrical  observations  on 
the  two  sides  of  the  apparatus,  this  would  be  no  source  of  discrepancy. 
The  effect  of  focussing  the  two  parts  of  the  grating  may,  in  the  first  instance, 
be  considered  as  a  prolongation  of  b'  till  it  cuts  x,  together  with  the  corre- 


IN    RELATION    TO    INTERFEROMETRY.  9 

spending  points  for  the  intersection  of  b"  with  x.  Thus  the  values  85'  and 
fa"  are  here  in  question  and  they  are 

£2'  =  i.97cm.  o2'  — ^,'  =  .05  cm. 

whence 

d2"  =  i.65  cm.  dl"=d2=.og  cm. 

are  the  conjugate  foci  for  the  extreme  rays  of  the  grating,  respectively, 
beyond  the  conjugate  focus  of  the  middle  or  normal  rays  b,  on  x.  Hence 
the  mean  of  the  extreme  rays  lies  at  .07  cm.  beyond  (greater  0)  the  normal 
ray,  and  the  X  found  in  the  first  instance  is  too  large  as  compared  with  the 
true  value  for  the  normal  ray. 

The  datum  .07  cm.  may  be  taken  as  the  excess  of  2X,  corresponding  to  the 
excess  of  angle  for  a  grating  one-half  as  wide  and  observed  on  both  sides 
(2*),  as  was  actually  the  case.  Finally,  since  not  the  whole  of  the  grating 
is  not  in  focus  at  once  a  correction  less  than  .07  cm.  for  2x  must  clearly  be 
in  question.  This  is  quite  below  the  difference  of  several  millimeters 
brought  out  in  §§4  and  6. 

To  make  this  point  additionally  sure  and  avoid  the  assumption  of  the 
last  paragraph,  we  will  compute  the  conjugate  focus  of  the  central  ray  (dif- 
ferent angles  6)  on  the  b'  focal  plane  parallel  to  the  grating  and  to  x  and  on 
the  b"  focal  plane  parallel  to  x.  The  computation  is  simpler  if  the  central 
ray  is  thus  focussed  than  if  the  extreme  rays  are  focussed  on  the  x  plane. 
The  distance  apart  will  be 

ds'  =  g  -  b'  cos  (0  +  0'}  (tan  (d  +  0'}  -  tan  6} 
ds»  =g-b"  cos  (6  -  6"}  (tan  0 -  tan  (0  -  6"} ) 

Inserting  the  results  for  0     0/     0/'     b'     b"    g 

o3'  =  .06  <V'=  ~~  -°4 

Both  the  b  foci  thus  correspond  to  large  angles.  Their  mean,  however, 
may  be  considered  as  vanishing  on  the  intermediate  x  plane. 

Thus  it  is  clear  that  the  effect  of  focussing  is  without  influence  on  the 
diffraction  angle  and  much  within  the  limits  of  observation.  It  is  therefore 
probable  that  the  residual  discrepancy  in  the  three  methods  is  referable  to 
a  lateral  motion  of  the  slit  itself,  due  to  insufficient  symmetry  of  the  slides 
A  A  and  BB  in  the  above  adjustment.  This  agrees,  moreover,  with  the 
residual  shift  observed  in  the  case  of  parallel  rays  in  §8.  The  remedy 
should  present  no  difficulty. 

10.  Reflecting  plate  grating.— The  adjustment  of  the  plane  grating  if 
cut  on  specular  metal  is  nearly  identical  to  the  above,  except  that  the  col- 
limator  is  fixed  as  a  whole  in  front  of  the  grating,  either  to  the  slide  carrying 
the  standard  of  the  grating,  B,  or  else  quite  in  front  of  the  cross  slide  A  A, 
fig. i  above,  so  as  to  give  clearance  for  the  to-and-fro  motion  of  the  rail,  R. 
This  admits  of  measurement  of  x  on  both  sides  of  the  slit,  so  that  2X,  the 
distance  apart  of  the  two  symmetrical  positions  for  a  given  spectrum  line, 
is  again  observed. 


10 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


II.  Rowland's  concave  grating.  —  For  the  case  of  the  concave  grating, 
the  accurate  adjustment  for  symmetrical  measurement  on  both  sides  of  the 
slit  is  not  feasible,  because  the  slit  and  eyepiece  would  have  to  pass  through 
each  other.  It  is  possible,  however,  to  find  conjugate  foci  at  different  dis- 
tances from  the  grating  in  the  normal  position,  which  approximately  answer 
the  purposes  of  measurement.  Rowland's  equation 

(cos  i/p—i/R)  cos  t+(cos  d/p'—i/p')  cos  6  =  0 

where  p  and  p'  are  the  conjugate  focal  distances  for  angles  of  incidence  and 
deviation  i  and  6,  may  for  0  =  o  be  written 


p/cos2  *  ~  R/cos  i  ~  |00  ~~  }f 
where  p0  is  the  normal  distance  of  the  eyepiece,  so  that 


P*   Po 

If  in  fig.  4  (page  5)  the  slit  S  is  put  at  p'>R  from  the  grating  G  (normal 
position),  the  image  is  at  E  at  the  end  of  p0  from  G,  where  p0<R<p'.  If 
po  be  used  as  a  rail  instead  of  R  and  put  at  an  angle  of  incidence  *',  for  the 
eyepiece  at  E'  or  E",  p0  cos  *  >  p.  But  this  excess  need  not  be  so  large  as 
to  interfere  with  adequately  sharp  focussing. 

Table  4  gives  an  example,  in  which  the  difference  of  p0  and  p0'  in  the  nor- 
mal position  is  even  over  i  foot,  an  excessive  amount,  as  the  distance  neces- 
sary for  clearance  need  not  be  more  than  a  few  inches.  The  grating  has 
14,436  lines  to  the  inch  and  a  radius  about  R  =  191  cm. 

TABLE  4. — Conjugate  foci  of  the  concave  grating.  7?  =  i9i  cm.,  14,436  lines  to 
inch,  5683  lines  to  cm.  D=. 000,176.  />„  =  166  cm.,  p  =  198  cm.,  p  —  /00  =  32  cm. 
i/pt—i/R  =  .000,788,  0=o,  sin  i  =  \/D. 


i° 

P 

(00  cos  i        Diff  . 

(/./cos  »)» 

Fraun- 
hofer             t 

lines,     j 

cm. 

cm. 

0° 

166.0 

166.0                .0 

27500 

B 

22°  59' 

5 

10 

165.3 

163  .2 

165.3                -o 
163.5                -3 

27500 
27400 

c 

D 

21°  54' 
19°  34' 

('5 

159-6 

160.3          -    -7         27300 

E 

17°  26' 

\ao 

I54-7 

156.0         —1.3         27100 

F 

I6°02' 

25 

148.5 

150.4          —i.  9         26800 

G           14°  10' 

3° 

140.9 

143.7          -2.8         26500 

35 

132.2 

136.0          —3-8         26000 

.... 

40 

122.3 

127.1          —4.8         25500 

The  greater  part  of  the  visible  spectrum  is  thus  contained  between  i  =  15° 
and  i  =  2o°.  It  follows  that  the  excess  of  p0  cos  i—p  lies  between  7  and  13 
mm.  Hence  the  eyepiece  may  be  placed  at  a  mean  position  corresponding 
to  10  mm.  and  give  very  good  definition  of  the  whole  spectrum  without 
refocussing,  as  I  found  by  actual  trial.  Within  i  cm.  the  focus  is  sharp 
enough  for  most  practical  purposes.  If  the  distances  p0  and  p0'  are  selected 
so  that  eyepiece  and  slit  just  clear  each  other  the  definition  is  quite  sharp. 


IN  RELATION  TO  INTERFEROMETRY.  11 

The  diffraction  equation  is  not  modified  and  if  2x  corresponds  to  the 
positions  -\-i  and  —i  for  the  same  spectrum  line, 


It  is  therefore  not  necessary  to  touch  the  eyepiece  and  this  is  contributory 
to  accuracy. 

If  Rowland's  equation  is  differentiated  relatively  to  p  and  p',—dp  = 

(P      V 
—  --  :  )  dp0'  where  the  factor  dp0'/p0'2  is  constant.      Hence  —  dp  varies 
Po  cos  ^/ 

as  (p/cos  *)2»  given  in  the  table.     If,  furthermore,  a  comparison  is  made 
between  dp0  and  dp  this  equation  reduces  to 

V  dpjdp  =  (R  —  p0(i  —  cos  i)  )/R  cos  z 
which  becomes  unity  either  for  i  =  o  or  for  p0=R  (Rowland's  case). 

12.  Summary.  —  By  using  two  slides  symmetrically  normal  to  each  other 
and  observing  on  both  sides  of  the  point  of  interference,  it  is  shown  that 
many  of  the  errors  are  eliminated  by  the  symmetrical  adjustments  in  ques- 
tion. The  slide  carrying  the  grating  may  be  provided  with  a  focussing  lens 
in  front  or  again  behind  it,  if  the  means  are  at  hand  for  actuating  the  slit 
which  is  not  sharply  in  focus  on  the  plane  of  the  eyepiece  carried  by  a 
second  slide  throughout  the  spectrum  at  a  given  time.  It  is  thus  best  to 
use  both  lenses  conjointly,  the  latter  as  a  collimator  and  the  former  as  an 
objective  of  the  telescope  in  connection  with  the  eyepiece.  It  is  shown  that 
a  centimeter  scale  parallel  to  the  eyepiece  slide  with  a  vernier  reading  to 
millimeters  is  sufficient  to  measure  the  wave-lengths  of  light  to  few  Ang- 
strom units  r  while  the  wave-lengths  are  throughout  strictly  proportional 
to  the  displacements  along  the  scale.  The  errors  of  the  three  available 
methods  and  their  counterparts  are  discussed  in  detail.  The  method  is 
applicable  both  to  the  transparent  and  the  reflecting  grating. 

It  is  furthermore  shown  that,  in  case  of  Rowland's  concave  grating,  obser- 
vation may  be  made  symmetrically  on  both  sides  of  the  slit,  by  providing  for 
reasonable  clearance  of  slit  and  eyepiece  passing  across  each  other,  although 
one  conjugate  focal  distance  is  now  not  quite  the  projection  of  the  other. 


CHAPTER  II. 


THE  INTERFERENCE  OF  THE  REFLECTED-DIFFRACTED  AND  THE  DIF- 
FRACTED-REFLECTED  RAYS  OF  A  PLANE  TRANSPARENT  GRATING, 
AND  ON  AN  INTERFEROMETER.  By  C.  Barus  and  M.  Barus. 

13.  Introductory. — If  parallel  light,  falling  on  the  front  face  of  a  trans- 
parent plane  grating,  is  observed  through  a  telescope  after  reflection  from 
a  rear  parallel  face  (see  fig.  5) ,  the  spectrum  is  frequently  found  to  be  inter- 
sected by  strong,  vertical  interference  bands.  Almost  any  type  of  grating 
will  suffice,  including  the  admirable  replicas  now  available,  like  those  of 
Mr.  Ives.  In  the  latter  case  one  would  be  inclined  to  refer  the  phenomenon 
to  the  film  and  give  it  no  further  consideration.  On  closer  inspection,  how- 


ever, it  appears  that  the  strongest  fringes  certainly  have  a  different  origin 
and  depend  essentially  on  the  reflecting  face  behind  the  grating.  If,  for 
instance,  this  face  is  blurred  by  attaching  a  piece  of  rough,  wet  paper,  or  by 
pasting  the  face  of  a  prism  upon  it  with  water,  so  as  to  remove  most  of  the 
reflected  light,  the  fringes  all  but  disappear.  If  a  metal  mirror  is  forced 
against  the  rear  glass  face,  whereby  a  half  wave-length  is  lost  at  the  mirror 
but  not  at  the  glass  face  in  contact,  the  fringes  are  impaired,  making  a 
rather  interesting  experiment.  With  homogeneous  light  the  fringes  of 

13 


14 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


the  film  itself  appear  to  the  naked  eye,  as  they  are  usually  very  large  by 
comparison. 

Granting  that  the  fringes  in  question  depend  upon  the  reflecting  surface 
behind  the  grating,  they  must  move  if  the  distance  between  them  is  varied. 
Consequently  a  phenomenon  so  easily  produced  and  controlled  is  of  much 
greater  interest  in  relation  to  micrometric  measurements  than  at  first 
appears  and  we  have  for  this  reason  given  it  detailed  treatment.  It  has 
the  great  advantage  of  not  needing  monochromatic  light,  of  being  appli- 
cable for  any  wave-length  whatever,  and  admitting  of  the  measurement  of 
horizontal  angles.  In  chapter  5  it  will  be  shown  that  these  interferences 
may  also  be  produced  by  Michelson's  or  by  Jamin's  apparatus. 


FIG.  6. — Showing  corresponding  triplets  of   diffracted  rays  for  a  single  incident 
ray,  and  each  of  the  cases  0,'>*  and  02'<t  of  fig.  5. 

When  the  phenomenon  as  a  whole  is  carefully  studied  it  is  found  to  be 
multiple  in  character.  In  each  order  of  spectrum  there  are  different  groups 
of  fringes  of  different  angular  sizes  and  usually  in  very  different  focal  planes. 
Some  of  these  are  associated  with  parallel  light,  others  with  divergent  or 
convergent  light,  so  that  a  telescope  is  essential  to  bring  out  the  successive 
groups  in  their  entirety.  At  any  deviation  the  diffracted  light  is  neces- 
sarily monochromatic,  but  the  fringes  need  not  and  rarely  do  appear  in 
focus  with  the  solar  spectrum.  If  the  slit  of  the  spectroscope  is  purposely 
slightly  inclined  to  the  lines  of  the  grating,  certain  of  the  fringes  may  appear 
inclined  in  one  way  and  others  in  the  opposite  way,  producing  a  cross  pat- 
tern like  a  pantograph.  The  reason  for  this  appears  in  the  equations. 

In  any  case  the  final  evidence  is  given  when  the  reflecting  face  behind 
the  grating  is  movable  parallel  to  it.  The  interferometer  so  obtained  is 


IN    RELATION    TO    INTERFEROMETRY. 


15 


subject  to  the  equation  (air  space  e,  wave-length  X,  angle  of  incidence  *',  of 
diffraction  #')•  5e  =  X/2(cos  6'—  cos  i),  and  is  therefore  less  unique  as  an 
absolute  instrument  than  Michelson's  classic  apparatus  or  the  device  of 
Fabry  and  Perot.  Its  sensitiveness  per  fringe,  5e,  depends  essentially  upon 
the  angle  of  incidence  and  diffraction  and  it  admits  of  but  i  cm.  (about)  of 
air-space  between  grating  face  and  mirror  before  the  fringes  become  too 
fine  to  be  available.  But  on  the  other  hand  it  does  not  require  monochro- 
matic light  (a  Welsbach  burner  suffices),  it  does  not  require  optical  plate 
glass,  it  is  sufficient  to  use  but  a  square  cm.  of  grating  film,  and  it  admits  of 
very  easy  manipulation,  for  painstaking  adjustments  as  to  normality,  etc., 
are  superfluous.  In  fact  it  is  only  necessary  to  put  the  sodium  lines  in  the 
spectrum  reflected  from  the  grating  and  from  the  mirror  into  coincidence 


FIG.  7. — Superposition  of  the  cases  of  figs.  5  and  6  for 


both  horizontally  and  vertically  with  the  usual  three  adjustment  screws  on 
grating  and  mirror.  Naturally  sunlight  is  here  desirable.  Thereupon  the 
fringes  will  usually  appear  and  may  be  sharply  adjusted  upon  a  second  trial 
at  once. 

When  the  air-space  is  small,  coarse  and  fine  fringes  (fluted  fringes)  are 
simultaneously  in  focus,  one  of  which  may  be  used  as  a  coarse  adjustment 
on  the  other.  Finally,  the  sensitiveness  per  fringe  to  be  obtained  is  easily 
a  length  of  one-half  wave-length  in  the  fine  fringes  and  one  wave-length  in 
the  coarse  fringes,  though  the  latter  may  also  be  increased  almost  to  the 
limit  of  the  former.  The  fine  fringes  compatibly  with  their  greater  sen- 
sitiveness vanish  within  about  a  millimeter.  Their  occurrence  with  the 
coarse  fringes  makes  it  possible  to  investigate  sudden  displacements  within 
the  latter. 


16  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


1. 

r^ 

I- 
II 


o   o 

11 

Ii 


is 

I 
I 


ouble 
ut  no 


s-c   ,v    £  7  r-a 
3§   £S    sj'J'g, 


J^p      C'EH         wT  ^    *S  "3     £  £  £  £ ' 
J  111  ll  II  -"SS: 


O«OO«^       «        O--  ...io 

00  00    ro  r^oo         rcoo    N     •      •         r>.    •     •     •  OO 

.^00  3 

U-J   U-)   1/1  IT)  IT)  IO   ' 

s 

«  :  2 

1 

o  usCJ"       i>4>:±!E'^       ~  J^^s  o  :    .         ««-! 

§          1  1.5  O       §ol3--o       b-«  S  '    " 

O       »ja.QS    ooa.Q>    >QZf 

:       :  :  :  :~TTTTrT^TT:       H     -T 

y...  &£•*'•  W'  ® 

^QCDC^      QCC^«^       -  H « — «  ^{ 

do  s  p 


.       • O  .00  .  .0 

VO-ss'rf1-  O  «.'   -   « 

A  ii  n     A  ii  ii        A  v  ii  n 


IN  RELATION  TO  INTERFEROMETRY.  17 

14.  Observations. — The  following  observations  were  made  merely  to 
corroborate  the  equations  used.    The  general  character  of  the  results  will 
become  clear  on  consulting  the  preceding  abbreviated  table  chosen  at  ran- 
dom from  many  similar  data.    An  Ives  replica  grating  with  15,000  lines  to 
the  inch  (film  between  plates  of  glass  .46  cm.  thick)  was  mounted  as  usual 
on  a  spectrometer  admitting  of  an  angular  measurement  within  one  min- 
ute of  arc.      Parallel  light  fell  on  the  grating,  fig.  5,  gg  (see  p.  13),  under 
different  angles  of  incidence,  i,  and  the  spectrum  lines  were  observed  by 
reflection  (after  reflection  from  gg  and  the  rear  face  ff)  at  an  angle  of 
diffraction  6'  in  air,  both  in  the  first  and  second  order  of  spectra,  and  so 
far  as  possible  on  both  sides  of  the  directly  reflected  beam.     In  view  of 
the  front  plate,  the  angle  i  corresponds  to  an  angle  of  refraction  r,  within 
the  glass  and  the  angle  6'  similarly  to  an  angle  of  diffraction  6,  respect- 
ively.   Hence  r>92  or  di<r  denote  the  sides  of  the  ordinary  ray  on  which 
observation  is  made.    As  a  rule  these  were  as  nearly  as  possible  in  the 
region  of  the  D  line  passing  toward  E.      Finally  50  denotes  the  angle 
between  two  consecutive  dark  fringes,  observed  and  computed  as  speci- 
fied.   Similarly  de  will  be  reserved  for  changes  of  thickness  e  of  the  glass 
and  8ef  for  changes  of  the  air  space  in  case  of  an  auxiliary  mirror  M. 

For  i  =  o°  the  number  of  groups  of  lines  was  a  single  set  in  each  order; 
but  only  the  end  of  the  spectrum  could  be  seen.  Measurements  refer 
(about)  to  the  C  line.  For  i  =  45°  several  groups  were  too  close  together,  or 
too  faint  for  measurement,  and  the  same  is  true  for  i=  22.5°.  An  estimate 
of  divergence  is  all  that  could  be  attempted  on  the  given  spectrometer. 
The  case  61  >r  was  usually  not  available,  but  for  2  =  22.5°  two  sets  were 
found  in  the  first  order,  one  being  the  normal  set.  The  fringes  in  all  cases 
decrease  in  size  from  red  to  violet,  but  less  rapidly  than  wave-length. 

Whether  they  are  convergent  or  divergent  for  a  given  set  of  fringes,  as 
for  instance  for  the  strong  set,  depends  on  the  position  of  the  grating.  Thus 
the  divergent  rays  become  convergent  when  the  grating  is  rotated  180° 
about  its  normal.  It  is  therefore  definitely  wedge-shaped.  In  fact  when 
the  auxiliary  mirror  M  is  used,  the  fringes  may  be  put  anywhere,  either  in 
front  of  or  behind  the  principal  focal  plane,  by  suitably  inclining  the  mirror. 

1 5.  Equations. — If  we  suppose  the  film  of  the  grating  gg  to  be  sandwiched 
in  between  plates  of  glass  each  of  thickness  e,  it  will  be  seen  that  triplicate 
rays  pass  in  the  direction  ti,  (0/>*)  or  of  fe,  (02'<O'),  which  will  necessarily 
produce  interference  either  partial  or  total.    With  respect  to  t\t  the  only 
light  received  comes  either  from  DI  by  direct  diffraction  at  gg,  or  from  RDi, 
by  reflection  from  the  lower  face/,  and  thereafter  by  diffraction  at  gg;  or 
from  DRi,  by  diffraction  at  gg  and  reflection  at  jf.    Similarly  the  light  along 
h  comes  in  like  manner  either  from  D2  or  DR2  or  RD^.    With  regard  to  the 
angles  of  incidence  and  refraction  or  of  diffraction  within  the  glass  or  out- 
side of  it,  we  have  the  equations  for  the  first  and  second  order  of  spectra 
(D  being  the  grating  space). 


18  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

sin  *'I  =  M  sin  r\  (i) 

sin  0i  '  =fji  sin  0i  (2) 

sin  Oz'  =  n  sin  02  (3) 

sin  r  —  sin  02  =  X/^MI  or  =  2\/Dn  (4) 

sin  0i  —  sin  r  =  X/.D/u,  or  =  2\/.D/*  (5) 

sin  *  —  sin  02'  =  \/D,  etc.  (4') 

sin  0/-sin  i  =  \/D,  etc.  (5') 

where  ju  is  the  index  of  refraction  of  the  glass,  found  to  be  equal  to  1.5265 
for  sodium  light,  by  breaking  off  a  small  corner  of  the  glass  of  the  grating 
and  using  Kohlrausch's  total  reflectometer. 

Furthermore,  for  the  occurrence  of  interference  along  t\  we  shall  have, 
for  the  case  of  the  three  rays  in  question,  respectively  combined  in  pairs, 
if  the  wave  fronts  be  taken  in  the  glass  plate  FFgg 

n\  =  2en  cos  r  (6*) 

nX=  2e/z(i  —  sin  0i  sin  r)/cos  0t  (7") 

n\  =  2en(  i  —  cos  (0i  —  r))/cos  0i  (8*) 


where  n  is  the  order  of  interference  and  a  whole  number,  e  the  thickness  of 
the  lower  glass  plate,  as  may  be  found  from  a  consideration  of  wave  fronts 
and  need  not  be  deduced  here.  It  will  be  seen  that,  whereas  equations  6 
and  7  present  cases  of  partial  interference,  equation  8,  which  is  virtually  the 
difference  of  6  and  7,  should  be  a  case  of  total  interference,  giving  strong 
fringes  and  possibly  useful  for  interferometry. 

Again  for  k  the  equations  will  be  similar  if  corresponding  wave  fronts  be 
taken  in  the  glass  plate  FFgg 

HA  =  2€jJ.  COS  r  (60 

nA=2ejji(i  —  sin  02sin  r)/cos  0,  (7') 

n/.  =  2eii(i  -  cos  (r  -  OJ  )/cos  0,  (80 

the  same  as  in  the  preceding  case  when  0i  is  replaced  by  02  and  0\—  r  by 
r—  02.  To  this  extent  there  is  no  essential  difference  between  the  cases. 
Therefore  0  may  be  used  indiscriminately  for  either. 

If,  however,  the  wave  fronts  be  taken  in  the  glass  plate  jffgg  the  equations 
become 

nA  =  2ejj.cosr  (6) 

n>l  =  2*/(  cos  0,  (7) 

HA  =  26/J.  (COS  T  —  COS  0|)  (8) 

with  three  other  corresponding  forms  for  0<r,  as  before.  The  latter  are 
the  true  equations  and  equation  8  is  the  case  of  total  interference. 

These  circumstances  are  at  first  quite  puzzling,  but  they  are  due  to  the 
fact  that  in  the  cases  6',  7',  8',  6",  7",  8",  the  wave  fronts  taken  correspond 
to  the  refracted  ray  only,  whereas  the  diffracted  rays  subsequently  undergo 


IN    RELATION    TO    INTERFEROMETRY.  19 

sudden  changes  of  deviation  on  leaving  gg.  We  have  nevertheless  carried 
both  groups  of  equations  6'  to  8'  and  6  to  8  in  mind,  though  the  latter  are 
alone  certain  of  application. 

For  an  air-space  between  gg  and  M  the  equations  would  be 

nh  =  2ecosi  nX  =  2ecos6f  nX=^  ie  (cos  6'  —  cos  i),  etc. 

16.  Differential  equations.— The  quantity  measured  on  the  spectrom- 
eter is  essentially  angular  and  preferably  dd'/dn,  the  angular  distance 
apart  of  the  fringes,  in  radians.  Later  we  shall  measure  5e  or  the  linear 
displacement  of  the  parallel  faces  per  fringe.  In  any  measurement,  how- 
ever, we  meet  with  embarrassment,  inasmuch  as  n,  X,  n,  r,  9,  9',  are  all 
variable.  The  angle  i  and  the  thickness  e  and  the  grating  space  D  are 
alone  given.  Among  these,  the  variation  of  r  with  /*  andX  must  be  found  by 
experiment.  Fortunately,  in  case  of  the  interferometer  all  these  variables 
are  eliminated  and  e  alone  changes,  subject  to  a  given  i  and  9'.  The  n  used 
need  not  be  known.  (See  paragraph  19.) 

For  the  present  purpose,  as  the  variation  of  p.  enters  only  as  a  correction, 
we  have  been  satisfied  with  the  usual  results  in  physical  tables.*  If  from 
the  C  to  the  D  line 


and  from  the  B  to  the  C  line,  =  —  .013,  we  may  write 

du.  dX 

-  ^  =  -OI57 

and  therefore 

/;\  di 

(9) 


We  shall  write  a  =.015,  b=  i+a. 

The  case  9>r  in  the  present  paper  is  not  of  much  experimental  interest, 
and  may  be  omitted  here.  For  the  case  of  r>9  we  shall  have  successively 
and  for  the  total  interferences  RD,  DR,  equation  8, 

dX  X  cos  0         dd 

—  -j—  =     .         j— : 2T    ~T~  (  I  I ) 

a/i      sin  r  —  b  sin  0    an 

dr  _      tan  r  cos  0      dO 
dn        sin  r  —  b  sin  6  dn 

dfi  _      afj.  cos  0        dO 
dn  ~s5Tr-&sin0   dn 

dd'  =  jfcos0(sinr-sin0)    dd 
dn  ~  cos  0'(sin  r-b  sin  6)  dn 

*  See  Kohlrausch's  Leitfaden,  nth  edition,  1910,  p.  712,  light  crown  glass  being 
taken. 


20  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

and  finally  corresponding  to  equations  6,  7,  8, 
dO^  _  _cosO     *  (sin  r  -sing) 
dn  ~  26  cos  0'    6^sin7sin0 

^  (sin  r-  sing) 


dn      26  cos  0'  6  cos  r  cos  0  +  a  sin  r  tan  r  cos  0 
d0'        cos  0  ^(sin  r  —  sin  0) 


dn     26  cos  0'  6( i  —  cos  (r  —  0))  +  a  sin  r  sin  0( i  —  cot  0  tan  r) 

the  last  term  in  the  denominator  being  corrective.     Here  dB'/dn  is  the 
observed  angular  deviation  of  two  consecutive  fringes,  so  that 
dn       dn  _  dn 

The  equation  corresponding  to  the  incorrect  equation  8'  would  have  been 
dd'  =     cos  0 
dn      26  cos  0' 

X  cos2  0(sin  r  —  sin  0) 
6cos20(i  —  cos(r  —  0))—  (sinr—  sin0)(sinr  —  6sin0j—  atan  rcos20sin(r  —  0) 

17.  Normal  incidence,  or  diffraction ,  etc. — For  the  case  of  normal  inci- 
dence i  =  r  =  o,  the  equations  corresponding  to  6,  7,  and  8  take  a  simplified 
form  and  are  respectively 

_  d00'  _  >1  sin  0  cos  0 

~~  dn        nha  ™°  "' 


_d00'_^sin0_ 
dn      26e  cos  0' 

d00'         cos  0         ^  sin  0 

—      -  —  ^s  — — —  — —  ( 1 8  ) 

dn      zbe  cos  0    i  —  cos  0 

If  0'  =  $  =  o,  for  normal  diffraction,  which  is  particularly  useful  in  Row- 
land's adjustments  as  well  as  on  the  spectrometer 

_  A  sin  r 

a-o      2e  ^C1*"008  r)~  a  sin  r  tan  r 

for  the  case  of  total  interference  corresponding  to  equations  8  and  17.    If 
j-=-0'orr=-0 

Wl         =  A  i 

_9     2^   tan  0  cos  0' 

18.  Comparison  of  the  equations  of  total  interference  with  observa- 
tion.— The  partial  interferences  corresponding  to  equations  6  and  7  are 
usually  too  fine  to  be  seen  unless  e  is  very  small.  They  amount  in  cases  of 
equations  15  and  16  for  e  =  .48  cm.  to  the  following  small  angles 

(iS)  (16) 

i=   o°            d6'/dn=.o6o'  dd'/dn=  .062' 

22.5°                        .048'  .050' 

45°                            -057'  -058' 


IN  RELATION  TO  INTERFEROMETRY.  21 

usually  less  than  four  seconds  of  arc  and  are  therefore  lost.  The  origin  of  the 
fine  interferences  actually  seen  in  the  table  is  thus  still  open  to  surmise. 
With  small  e  and  the  interferometer  they  are  obvious. 

The  total  interferences  as  computed  in  the  above  table  agree  with  the 
observations  to  much  within  o.  i  minute  of  arc  and  these  are  experimental 
errors;  particularly  so  as  it  was  not  possible  to  use  both  verniers  of  the 
spectrometer.  The  interesting  feature  of  the  experiment  and  calculation 
is  this,  that  86'  has  about  the  same  value  for  all  incidences  i  from  o°  to  45° 
and  even  beyond.  The  equations  do  not  show  this  at  once,  owing  to  the 
entrance  of  /*  and  r.  But  apart  from  a  and  6  equation  17  is  nearly 


_ 
dn     2efjL    i-cos(r-0)  W 

which  is  independent  of  r  to  the  extent  in  which  cos  (r  —  0)  is  constant.  The 
dependence  of  dO'/dn  on  wave-length  is  borne  out.  (See  paragraph  19.) 
Finally,  dd'  '  /dn  is  independent  of  /*  except  as  it  occurs  in  a  and  b. 

If  the  glass  plate  JJgg  is  removed  and  a  mirror  M  used,  as  in  the  inter- 
ferometer, the  fringes  may  be  enormously  enlarged  by  decreasing  e  and  the 
measurements  made  with  any  degree  of  accuracy;  but  such  measurements 
were  originally  impracticable  and  have  little  further  interest  in  this  place 
since  the  interferometer  itself  is  tested  in  the  next  chapter. 

19.  Interferometer.—  The  final  test  of  the  above  equation  is  given  by  the 
last  part  of  the  table  for  different  thicknesses  of  glass,  e  =  .48  and  e=  .77  cm. 
The  results  are  in  perfect  accord. 

These  data  suffice  to  state  the  outlook  for  the  interferometer.  In  this 
case  n  and  e  are  the  only  variables,  so  that  equation  8  becomes 

de  =  >l/2//(cos  0  -  cos  r)  (20') 

where  8e  is  the  thickness  of  glass  corresponding  to  the  passage  of  one  fringe 
across  the  cross-hairs  of  the  telescope  or  a  definite  spectrum  line. 

If  instead  of  glass  in  the  grating  above,  an  air-space  intervenes  between 
the  film  of  the  grating  and  the  auxiliary  mirror  M,  fig.  5,  the  equation 
reduces  to 

dg  =  _  ;__     _  =  __  J  _ 
2  (cos  6'  -  cos  *)     2  (  ]/  i-(sint-J/Z?)*  -  cos  i) 

where  i  and  6'  are  the  angles  of  incidence  and  diffraction  in  air. 

These  equations  20  embody  a  curious  circumstance.  Inasmuch  as  B  and 
6'  change  as  *  increases  from  o°  to  90°  from  negative  to  positive  values  at 
about  i=  13°  and  1  =  20°,  respectively,  the  denominator  of  either  equation 
20  will  pass  through  zero  (for  air  at  about  i=  10°).  Hence  at  this  value  of 
*  the  motion  of  the  mirror  M  produces  no  e  effect  (stationary  fringes), 
while  on  either  side  of  it  the  fringes  travel  in  opposite  directions  in  the 
telescope  when  e  changes  by  the  same  amount.  In  the  negative  case  the 
sensitiveness  for  air-spaces  passes  from  8e——  .000489  to  de=—  <*>,  per 


22  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

fringe.  In  the  positive  case  from  5e=  +  *>  to  8e=.ooo  039,  per  fringe  or 
to  a  limit  of  about  a  half  wave-length  in  case  of  15,000  lines  to  the  inch. 
This  limiting  sensitiveness  may  be  regarded  as  practically  reached  even  at 
1  =  40°  where  8e=.ooo  155  cm.  per  fringe  and  an  angle  of  about  1  =  45°  is 
most  convenient  in  practice. 

In  addition  to  the  large  fringes  the  fine  set  appears  when  e  is  small  or  not 
more  than  a  few  tenths  of  a  millimeter.  The  sensitiveness  of  these  is 
naturally  much  more  marked.  In  the  two  cases 

de  =  h/2  cos  i  (20') 

de  =  1/2  COStf  (20") 

so  that  nearly  A/ 2  per  fringe  is  easily  attained,  but  the  available  thickness 
of  air-space  within  which  they  are  visible  is  decreased. 

At  *  =  20°  about,  and  in  case  of  an  air-space  6'  is  nearly  o°.  We  suggested 
above  that  these  fine  fringes  may  be  used  as  a  fine  adjustment  in  connection 
with  the  large  fringes,  on  which  they  are  superimposed.  In  appearance 
these  large  fluted  fringes  are  exceedingly  beautiful.  The  fine  fringes  have 
the  limiting  sensitiveness  of  the  coarse  fringes,  i.e.,  the  cases  for  *  =  go° 
or  0'  equal  to  maximum  value.  In  different  focal  planes  both  sets  of  fine 
fringes  may  be  seen  separately  for  small  e  (air  wedge) . 

Equation  20  shows  that  for  smaller  grating  spaces  D,  and  consequently 
also  in  the  second  order  of  spectra  there  must  be  greater  sensitiveness, 
caet.  par. ;  but  as  a  rule  we  have  not  found  these  fringes  as  sharp  and  useful 
as  those  in  the  first  order. 

The  limiting  sensitiveness  per  fringe,  however,  follows  a  very  curious 
rule.  If  in  equation  20  we  put  t  =  go°, 


zde=  1   W/(2-l/D)  =l/Vr(2-r)  in  the  first  order 
if  r=\/D,  and 


2de=  \/W/4(i—A/D)  =  XJ2\  r(i  —r)  in  the  second  order 

D  is  the  grating  space.  Both  equations  have  a  minimum,  8e  =A/2 ,  at  \/D  =  i 
in  the  first  order  and  A/D  =  .5  in  the  second  order,  beyond  which  it  would  be 
disadvantageous  to  decrease  the  grating  space.  These  minimum  conditions 
are  as  good  as  reached  even  when  D  corresponds  to  15,000  lines  to  the  inch, 
as  above,  where  roughly  io68e  =  $&  cm.  in  the  first  order  and  io65e  =  33  cm. 
in  the  second  order. 

All  the  conditions  discussed  above  are  summarized  in  fig.  8  and  fig.  9 
for  the  first  and  second  orders  of  spectrum. 

To  view  the  stationary  fringes  of  the  first  order  was  not  practicable, 
since  they  occurred  for  i=  10°,  whereas  the  telescopes  were  in  contact  at 
about  20°.  In  the  second  order  of  spectra  they  may  be  approached  more 
nearly  as  they  occur  when  *  is  roughly  20°.  If  the  distance  e  is  made  small 
enough,  so  that  the  three  cases  of  equations  20,  20',  20*,  are  visible,  the 
appearance  is  very  peculiar.  The  fringes  of  equation  20  are  very  slow- 


IN    RELATION    TO    INTERFEROMETRY. 


23 


moving.  They  are  intersected  by  the  small  fringes  of  equation  20',  pro- 
ducing the  fluted  pattern  already  discussed.  Over  all  travel  the  rapidly 
moving  fringes  of  equation  20",  producing  a  kind  of  alternation  or  flickering 
which  it  is  very  difficult  to  analyze  or  interpret  until  e  is  very  small,when 
all  three  sets  are  broad  and  easily  recognized.  Sunlight  should  be  used. 
Nothing  like  these  alternating  fringes  was  seen  in  the  first  order,  but  we 
used  a  film  grating  only. 


FIG.  8. — Charts  showing  dependence  of  0  on  r,  0'  on  i,  cos  0'  and  cos  i  on 
i,  8e  on  r  and  de  on  i,  de  on  X/D  for  the  first  order  of  spectrum. 

The  above  equation  shows  finally  that  de  is  not  exactly  proportional  to 
wave-length,  though  the  former  decreases  with  the  latter,  as  shown  above. 

The  three  equations  20  indicate  finally  that  for  i>6'  all  fringes  travel 
in  the  same  direction  with  increasing  e;  whereas  if  6'  >  i,  the  set  correspond- 
ing to  equation  20  travel  in  a  direction  opposite  to  that  of  the  sets  20'  and  20". 


24 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


This  is  strikingly  borne  out  by  making  the  experiment  for  6>i  with  a  small 
angle  «,  both  in  the  first  and  second  order. 

Table  6  contains  a  few  data  obtained  by  carrying  the  mirror  on  a  Fraun- 
hofer  micrometer,  reading  to  .0001  cm.,  toward  a  stationary  grating  film. 


0°      10°     20°      30°     40°      50°     60°     70° 


FIG.  o. — Similar  data  to  those  of  fig.  8,  for  the  second  order  of  spectrum. 
In  addition,  dd'/dn  is  shown  as  given  by  equations  15,  16,  17. 

Observations  were  made  in  the  region  of  the  D  lines.  The  grating  was 
originally  between  plates  of  glass  £=.48  cm.  thick.  Finally  the  plate 
between  grating  and  mirror  was  removed,  the  whole  distance  now  being  an 
air-space.  This  has  no  effect  on  8e,  but  e  may  then  be  reduced  to  zero  and 
the  fringes  enlarged. 


IN    RELATION    TO    INTERFEROMETRY. 


25 


These  data  merely  test  the  equations,  as  no  special  pains  were  taken  for 
accurate  measurement,  which  neither  the  micrometer  screw  nor  the  special 
adjustments  warranted.  Usually  the  micrometer  equivalent  of  50  fringes 
was  observed  on  the  screw.  The  maximum  distance  e  between  grating  and 
mirror  was  .48  cm.  of  glass  and  .25  cm.  of  air  conjointly,  or  within  i  cm. 
In  case  of  fine  fringes  mere  pressure  on  table  or  screw  impaired  the  adjust- 
ment. Moreover  these  fine  fringes  run  through  the  shadow  of  the  coarse 
fringes,  and  their  size  in  consecutive  spaces  between  the  latter  seems  to  vary 
periodically  as  if  they  alternated  between  the  two  equations  15  and  16. 

TABLE  6. — Interferometer  measurements.     Replica  grating.      Air-space,  often  in 
addition  to  glass  space,  e  =  .46  cm. 


' 

e' 

Coarse  fringes. 

Fine  fringes. 

Air-space.* 

Observed. 
SeXio* 

Computed. 
SeXio* 

cm. 

Co.,  391 

130 
68 
324 
240 

Air-space. 

Observed. 
teXio* 

Computed. 
SeXio* 

22°  30' 

45       o 
67    30 
25       o 

30          2 

2°      6' 
21         I 

35       i 
4      7 
8    37 

cm. 
.02  to    .33 

.013  to  .250 

cm. 
39° 

72 
322 
241 

cm. 

cm. 

cm. 

Glass  removed  between  mirror  and  grating. 

32°  25' 

i°4o' 

.013  to  .054 

387 

392 

.006  to  .025 

34 

(32 
13° 

45      o 

20    49 

.032  to  .064 

129                 130 

.000  to  .007 

33              {32 

*  Approximate;  contact  endangering  adjustment. 

20.  Secondary  interferences. — We  come  now  to  consider  the  minor 
interferences  which  are  either  weaker,  finer,  or  more  diffuse  than  the  strong 
forms  discussed.  In  the  interpretation  of  these  we  have  not  met  with  suc- 
cess, but  some  reference  to  them  is  essential.  We  assume  that  after  two 
reflections  the  fringes  can  no  longer  be  seen. 

In  fig.  6  if  there  is  light  reinforcement  passing  in  any  direction  t\  or  h\ 
then  each  incident  ray  /,  at  an  angle  i  with  the  normal  n,  will  after  refrac- 
tion be  represented  by  the  six  rays  A  RDi  DRi  (ft  > r) ,  A  DRZ  RD2  (02<r), 
as  shown  in  the  figure,  under  a  notation  similar  to  the  preceding  case.  If 
these  rays  are  brought  to  a  focus  by  a  telescope  there  must  be  interferences 
between  pairs  of  which  the  fringes  of  D,  RD,  and  D,  DR,  will  usually  be 
too  fine  to  be  visible,  whereas  RD,  and  DR,  will  be  large  enough  to  be  seen. 

As  the  lines  are  not  quite  sharp,  measurement  is  difficult  and  no  other 
fringes  were  therefore  computed,  and  we  have  not  thus  far  been  at  pains  to 
discover  a  reason  for  the  large  fringes  in  table  i,  either  in  the  first  order 
(*  =  22.5°,  60' =  4. 7')  or  in  the  second  order  (4  =  45°,  50' =  3.3')- 


26  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

21.  Summary  of  secondary  interferences.— The  possible  partial  inter- 
ferences are  naturally  numerous.     If  we  superimpose  the  case  of  fig.  5 
on  fig.  6  and  draw  all  the  rays  of  the  first  reflection  for  6>r  only  (to  avoid 
complications),  i.  e.,  for  a  single  direction,  it  will  be  seen  that  nine  rays  are 
included,  of  which  a,  b,  c,  a',  b',  c',  a",  b",  c",  come  from  a  single  incident  ray 
each.    Thus  we  can  assemble  these  interferences  from  a  determinant  like 

a  b  c 
a'  b'  c' 
a"  b"  c" 

and  there  should  be  18  cases.  Most  of  these  are  identical  in  path  difference, 
but  they  have  not  led  us  to  any  satisfactory  identification.  They  may 
therefore  be  omitted  here. 

Other  difficulties  enter .  Fringes  which  may  be  invisible  if  observed  for  the 
glass  plate  may  be  visible  if  arising  in  the  collodion  film,  as  this  is  very  thin. 
Possibly  some  of  the  finer  lines  may  arise  in  this  way,  but  not  probably. 

The  tendency  of  certain  groups  of  interferences  to  travel  in  opposed 
directions  with  rotation  of  the  slit  has  suggested  to  us  the  possible  occur- 
rence of  zone-plate  action,  where  there  would  be  multiple  foci.  But  we 
have  not  succeeded  in  establishing  coincidences  either  for  the  virtual  or 
real  foci  in  such  a  case.  Moreover  this  opposition  in  motion  has  already 
been  accounted  for  in  paragraph  19. 

22.  Convergent  and  divergent  rays.— What  finally  characterizes  the 
above  groups  of  interferences  is  the  difference  in  position  of  their  focal 
planes.    They  rarely  coincide  with  the  spectrum  (parallel  rays)  and  hence 
do  not  always  destroy  it.    If  present  with  the  spectrum  the  latter  is  wholly 
wiped  out.    If  the  strong  fringes  are  convergent  for  a  given  adjustment  of 
grating  they  become  divergent  when  the  grating  is  rotated  180°  about  its 
normal.    Hence  the  plates  of  glass  are  sharply  wedge-shaped  and  to  these 
differences  the  location  of  focal  planes  is  to  be  referred. 

In  addition  to  this  the  three  regular  reflections  are  not  in  the  same  focus 
which  shows  the  surfaces  (collodium  film)  to  be  slightly  curved.  The  above 
experiments  succeed  best  when  two  of  the  reflections  are  yellowish,  which 
probably  means  that  the  grating  face  is  from  the  observer. 

Suppose  the  remote  glass  face  makes  an  angle  dr/2  with  the  surface  of  the 
grating.  Then  the  DR  ray  of  the  strong  interferences  has  its  angle  incre- 
mented by  d6=dr,  whereas  the  RD  ray  receives  an  increment  of  but 

dd  = -Q  dr.  Hence  if  the  DR  and  RD  rays  were  parallel  for  parallel  sur- 
faces they  would  be  at  an  angle  corresponding  to 

d6  —  dr     cosr  —  cos  0 

~dT       "7os0 

where  dr/2  is  the  angle  of  the  wedge.  Thus  for  the  partial  case  of  single 
incidence,  fig.  6,  dB>dr  if  6>r,  or  the  issuing  rays  would  converge;  and 
d0<dr  and  r>  0,  or  the  issuing  rays  would  diverge.  If  DR  is  negative  the 


IN    RELATION    TO    INTERFEROMETRY.  27 

opposite  conditions  will  hold,  since  dr  and  d6  change  signs  together.  For 
the  case  of  triple  incidence,  fig.  5,  there  will  be  similar  relations  with  less 
liability  to  convergence.  The  interferences  are  further  modified  by  the 
change  of  thickness  of  glass  or  the  variable  e  implied. 

Fig.  7  shows  that  rays  all  but  parallel  will  cross  each  other  in  front  (con- 
vergent) or  behind  (divergent)  the  grating,  depending  on  their  mutual  lat- 
eral positions.  As  a  ray  moves  from  the  right  to  the  left  of  the  normal,  the 
phenomenon  may  change  from  divergence  to  convergence  and  vice  versa. 

23.  Measurement  of  small  horizontal  angles.  —  These  relations  are  very 
well  brought  out  by  the  interferometer,  in  which  the  mirror  M  may  be  in- 
clined at  pleasure.  If  small  values  of  deviation  only  are  in  question,  this 
instrument  becomes  a  means  of  measuring  small  horizontal  angles  7  be- 
tween mirror  and  grating  as  these  involve  less  change  of  focus.  In  fact  if 
h  is  the  vertical  height  of  the  illumination  at  the  mirror  M  and  the  cor- 
responding obliquity  of  fringes  is  equivalent  to  an  excess  of  N  fringes 
crossing  the  bottom  of  the  cross-hairs  as  compared  with  the  top  for  a 
wave-length  X,  7  =  N8e/h;  or 


The  question  next  at  issue  is  thus  the  value  of  h.  It  will  be  noticed  that 
if  parallel  rays  fall  upon  the  slit,  they  will  be  brought  to  a  focus  by  the  col- 
limator  objective  first,  and  thereafter  by  the  telescope  objective,  placed  at 
a  diametral  distance  D  beyond  it.  Then  if  5  is  the  vertical  length  of  slit 
used,  and  /0  and  /,  the  focal  lengths  of  the  two  objectives,  respectively,  it 
follows  that  the  length  h  =  5  is  virtually  illuminated.  Hence, 

NX 
7  ~~  2S  (cos  0'-  cos  *) 

For  since  the  angle  7,  or  a  ratio,  is  in  question,  N8e/h  is  constant  and  it 
makes  no  difference  where  the  mirror  M  may  be  placed,  i.e.,  how  great  the 
absolute  vertical  height  of  the  illumination  h  may  be. 

In  case  of  this  method  (parallel  light  impinging  on  the  slit)  the  illumin- 
ation at  each  point  of  the  image  is  received  from  but  a  single  point  (nearly) 
of  the  mirror,  whereas  if  the  light  falling  on  the  slit  is  convergent,  the  whole 
vertical  extent  of  the  mirror  illumination  contributes  to  each  point  of  the 
image  in  the  ocular.  Hence  in  the  latter  case  the  fringes  are  only  sharp 
when  M  and  the  grating  are  rigorously  parallel,  and  they  soon  become 
blurred  when  this  is  increasingly  less  true.  The  same  observation  also 
accounts  for  the  greater  difficulty  in  adjustment  when  lamp-light  is  used. 
In  any  case,  equation  2  2  furnishes  N/S.  N  may  be  obtained  with  an  ocular 
micrometer. 

In  the  interest  of  greater  precision  the  angle  7  may  be  found  by  actually 
measuring  the  inclination  to  the  vertical  /3  of  the  fringes  in  the  ocular. 
Here  if  the  height  of  image  5  in  the  ocular  corresponds  to  the  vertical  length 
of  slit  5, 


28  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

H-K) 

while 

v,,    ,;«' 

5        dn 
where  dd'/dn  is  given  by  equation  1  7  .    Hence  s  may  be  eliminated  and 

(24> 


.  ;. 

If  now  we  further  eliminate  Ar/5  in  equation  22  by  equation  24,  we  have 
finally 


'      a/eft(cos  0'  -  cos  i)dO'/dn 

so  that  7  is  given  in  terms  of  ft,  the  observed  inclination  of  fringes  in  the 
ocular.  To  measure  ft  the  ocular  must  be  revolvable  on  its  axis,  so  that  the 
cross-hairs  may  be  brought  into  coincidence  with  the  fringes  and  the  angles 
may  be  found.  To  measure  M,  the  D  lines  as  they  remain  vertical  may 
often  be  used,  if  in  focus,  for  reference  in  place  of  vertical  cross-hairs. 
Using  the  d  ata  of  the  above  experiments,  if  t  =  45°— 

N  =  i  /C=/,  =  Z)  (nearly)  =23  cm.,  cos  0'—  cos  i=  .2264 

S  =  .9cm.  >l  =  6oXio-6  <f0'/dn  =  493  Xio~6 

whence  7  =  146  X  lo"6  radians,  or  about  a  half-minute  of  arc  per  fringe,  and 
0  =  44'  per  fringe.  Thus  ft  is  about  88  times  as  large  as  7.  At  i  =22.5°, 
7=  1.5'  per  fringe,  ft  =  45'  per  fringe.  Naturally  the  sensitiveness  increases 
with  the  angle  of  incidence  *.  When  the  fringes  are  large,  one-tenth  fringe 
is  easily  estimated,  so  that  a  horizontal  angle  7  of  a  few  seconds  between 
mirror  and  grating  should  be  measurable.  An  ocular  micrometer,  as  sug- 
gested, would  carry  the  precision  beyond  this. 

24.  Summary.  —  It  has  been  shown  that  the  interferences  here  in  question 
take  place  in  accordance  with  the  equations  (6),  (7),  and  (8)  above.  They 
therefore  occur  in  triplicate,  but  not  necessarily  in  the  same  focus.  Their 
superposition  gives  rise  to  fluted  interference  patterns  of  which  the  coarse 
fringes  (8)  are  less  sensitive  to  micrometer  displacements  than  the  fine 
fringes  (6)  and  (7),  on  the  one  hand,  while  they  persist  over  a  comparatively 
greater  range  of  path  difference,  on  the  other.  The  eventual  sensitiveness 
of  both  approaches  the  wave-length  of  light.  The  adjustment  made  is 
essentially  self-compensating,  in  a  way  that  excludes  the  wedge-angle  of  the 
plate;  and  the  lines  are  rigorously  straight  and  vertical,  to  the  extent  in 
which  they  maybe  regarded  as  parts  of  the  periphery  of  ellipses,  whose  centers 
are  infinitely  distant  on  the  same  horizontal.  As  a  whole,  therefore,  these 
interferences  are  special  cases  of  the  phenomenon  described  in  chapters  IV 
and  V,  where  they  may  be  made  to  pass  through  the  zero  of  air-space. 

The  lines  are  inclined  for  a  wedge-shaped  air-space  in  a  manner  admitting 
of  the  measurement  of  very  small  angles. 


CHAPTER   III. 


THE  GRATING  INTERFEROMETER.     By  C.  Barus  and  M.  Barus. 

25.  Introductory. — In  view  of  the  perfection  which  has  been  attained 
in  the  construction  of  film  gratings  and  of  the  simplicity  of  the  instru- 
mental equipment  needed,  we  have  been  at  some  pains  to  put  the  type  of 
interferometer  recently  described*  into  practical  form.  This  is  shown  in 
the  accompanying  diagram,  in  which  fig.  10  is  a  photographic  view,  the 
parts  of  which  are  easily  recognized  by  aid  of  the  simplified  plan  and  ele- 
vation in  fig.  n,  A  and  B. 


<£*£  o 


FIG.  ii. — A.  The  same  (fig.  10)  in  plan.     B.  The  same  in  elevation. 

The  attachment  for  magneto-striction  is  purely  incidental.  The  small 
increments  of  length  in  question  were  thought  to  offer  an  excellent  test  of 
the  availability  of  the  interferometer  for  micrometric  length-measurements 
when  these  are  of  the  sudden  type  and  unlike  the  regular  expansions. 


*  Science,  xxxi,  p.  394,  1910;  Phil.  Mag.  (6),  xx,  p.  45,  1910;  above  chapter  II. 

29 


30  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

26.  Apparatus.— In  fig. n,  A  is  the  collimator,  B  the  telescope  for  view- 
ing the  interference  patterns  reflected  from  the  grating  gg  and  mirror  M 
over  the  revolvable  table  CC.  These  are  the  essential  parts.  The  remain- 
der is  the  incidental  mounting  just  referred  to,  O  being  the  helix  containing 
an  iron  rod,  soldered  at  its  ends  to  brass  or  copper  tubes,  /  and  tr,  the  latter 
very  light  and  movable  in  the  V-groove  N,  the  former  /  heavier  and 
clamped  in  the  upright  P. 

The  whole  is  supported  on  a  frame  of  quarter-inch  gas  piping  and  con- 
nections uFvR,  the  feet  being  at  uvR,  only  the  latter  appearing  in  the  figure. 
Under  the  grating  is  an  attachment  *  with  four  or  eight  screw  sockets,  W, 
respectively  at  right  angles  to  each  other,  into  which  a  variety  of  apparatus 
may  be  screwed,  in  the  absence  of  the  removable  pipe  pp.  When  p  is  not 
used  it  is  replaced  by  a  foot. 

Telescope  B  and  collimator  A  are  carried  on  T-pieces,  with  nipples  at 
the  same  height  (about  8  cm.  or  less)  above  the  iron  frame,  and  B  may  be 
slightly  inclined  about  a  horizontal  axis.  It  may  also  be  provided  with  a 
micrometer  eyepiece,  as  shown  in  fig.  10. 

The  table,  CC,  carrying  the  grating,  is  revolvable  on  an  upright  H  and 
may  be  clamped  to  fixed  verniers,  D,  also  carried  at  d  by  the  upright  H. 
The  table  is  cut  away  on  the  further  side  so  that  vertical  apparatus  may 
approach  closer  to  the  grating  in  GG.  It  would  even  be  an  advantage  to 
use  (not  much  more  than)  a  semicircular  table  with  verniers  DD  alter- 
nately in  use,  set  at  an  angle. 

On  the  table  two  parallel  brass  guides  determine  the  motion  of  the  slide, 
rr,  actuated  by  the  micrometer  screw  E. 

The  grating  is  mounted  with  the  film  on  a  glass  plate  gg,  and  exposed  on 
the  side  away  from  the  observer,  facing  the  mirror  M .  It  is  adjusted  in  a 
rectangular  shallow  brass  case,  GG,  with  the  side  toward  M  and  the  top 
open.  The  figure  shows  three  adjustment  screws  actuating  the  rear  of  the 
glass  plate,  gg,  and  the  springs  ss,  which  push  the  plate  to  the  rear,  passing 
through  slits  in  the  sides  of  GG.  There  is  one  spring  for  the  top  and  one  for 
the  bottom  of  the  grating.  The  capsule  GG  is  firmly  attached  to  the  slide 
rr,  so  that  the  grating  may  be  moved  fore  and  aft  by  the  micrometer  screw. 

For  some  purposes  the  mirror  M,  similarly  adjustable  by  three  screws, 
may  be  attached  to  the  plate  CC,  free  from  the  slide  rr.  In  the  figure  the 
mirror  is  on  the  light  brass  tube  t',  which  makes  a  prolongation  of  the  iron 
rod,  here  to  be  tested  for  magneto-striction.  The  silvered  face  of  M  fronts 
the  grating. 

The  coil,  O,  used  is  wound  on  a  slender  annular  chamber  or  double  tube, 
through  which  cold  water  may  be  kept  in  circulation  (see  fig.  10),  to  obviate 
changes  of  temperature  of  the  rod.  The  uprights  P,  T,  T,  N,  in  figs.  10 
and  ii,  moreover,  are  adjustable  up  and  down  as  well  as  around  the  rod 
pp,  by  clamps  shown  in  fig.  10,  in  which  other  unessential  details  may  be 
seen,  including  the  hose  for  supplying  and  removing  water. 

*  In  a  later  form  a  special  base  R'  was  provided  below  \V. 


IN    RELATION    TO    INTERFEROMETRY.  31 

27.  Adjustments. — The  micrometer  screw  controlled  at  E  has  a  double 
purpose.  In  the  first  place  it  is  an  immediate  check  on  all  measurements 
of  length  increment  made.  It  shows,  moreover,  whether  the  displacements 
are  expansions  or  contractions,  since  it  may  compensate  any  motion  of  the 
mirror.  In  the  second  place,  since  the  collimator  A  and  telescope  B  are 
clamped  at  a  fixed  distance  apart  approximately,  it  enables  the  observer 
to  put  the  part  of  the  spectrum  needed  into  the  center  of  the  field.  For  the 
angle  aMc  may  here  be  considerably  increased  or  decreased,  and  both 
telescopes  are  revolvable  about  the  vertical.  They  may  also  be  raised 
or  lowered,  moved  right  and  left  and  rotated  about  the  horizontal,  being 
grasped  by  an  ordinary  clamp  (not  shown).  The  head  E  of  the  screw, 
moreover,  as  well  as  the  adjustment  screws  of  the  grating,  are  at  all  times 
within  easy  reach  of  the  observer.  After  an  initial  rough  adjustment  (the 
direct  reflections  from  mirror  and  grating  being  put  into  coincidence),  the 
adjustment  screws  on  the  grating  are  manipulated  until  the  three  spectra 
seen  in  the  telescope  B  coincide  both  horizontally  and  vertically.  For  this 
purpose  the  D  and  E  lines  of  the  spectrum  are  useful  and  sunlight  is  prefer- 
able. In  the  absence  of  sunlight  a  small  electric  arc  lamp  with  the  rays 
issuing  nearly  parallel  suffices  equally  well,  supposing  the  apparatus  is  in 
adjustment.  When  telescope  and  collimator  are  fixed,  motion  at  the 
micrometer  screw  naturally  does  not  displace  the  spectrum  line  X,  on 
which  the  telescope  has  been  focussed,  so  that  the  interferences  recorded 
pass  through  a  definite  part  of  the  spectrum. 

The  three  sets  of  interferences  available  are  subject  to  the  equation 

de" '  =  XJ2  cos  i  (i) 

de'=Xl2COsd  (2) 

de  =  A/2(cosO  —  cosi)  (3) 

where  X  is  the  wave-length  of  light  used,  i  and  B  the  angles  of  incidence  and 
of  diffraction  in  air,  and  where  be,  be' ,  de",  represent  respectively  the  incre- 
ment of  air-space  between  mirror  and  grating  per  fringe  passing  the  cross- 
hairs. It  is  therefore  necessary  to  know  the  angles  6  and  i,  and  for  this 
purpose  the  verniers  D  and  the  revolvable  plate  CC  graduated  on  its  edge 
are  provided. 

Let  the  mirror  or  grating  plate  be  turned  until  the  reflected  image  of  the 
slit  coincides  with  the  slit  itself.  Here  the  observer  must  be  able  to  look 
at  the  inner  face  of  the  slit  in  A  through  the  hole  h  in  the  tube.  Then  let 
the  angle  be  read  off.  Thereafter  let  the  plate  C  be  undamped  and  turned 
until  the  slit  is  seen  sharply  on  the  cross  wire  of  the  telescope  C  and  a  second 
reading  made.  The  angle  so  observed  is  (*-f-0)/2=a.  We  may  therefore 
write 

sin  i  — sin  0=2  cos  (t+0)/2  •  sin  (i  —  6)/2=A/D 

where  D  is  the  grating  space.     Thus 


32  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

from  which  (i—d)/2=b  may  be  found.  Hence  i  =  a+b;  B=a—b.  An 
eccentric  position  of  the  grating  is  of  no  consequence. 

In  the  given  apparatus  the  inner  angle  of  diffraction  has  been  utilized. 
This  brings  A  and  B  closer  together  for  a  sufficiently  large  angle  i  to  secure 
the  best  results. 

In  a  later  construction  of  the  apparatus  on  an  independent  foot  under  W, 
uv  is  a  long  smooth  rod  and  A  and  B  are  attached  by  clamps  admitting  of 
motion  right  and  left,  up  and  down,  and  rotation  about  the  horizontal 
and  vertical. 

28.  Angular  extent  of  the  fringes. — In  chapter  II  the  equations  were 
worked  out  (I.e.)  for  the  more  complicated  case  of  a  medium  of  thickness  e 
and  refractive  index  p.  For  the  case  of  an  air-space  the  equations  become 
much  simplified.  Corresponding  to  equations  i,  2,  3,  if  6<i  and  sin  *—  sin 
B=\/D 

dO'  =   /.-  i 

dn      2eD  cos  /  cos  6 

dn      2eD   i  —  sin  i  sin  0 

dd  =   *  -— -  (6) 

dn     2eD   i  —  cos  (i  —  0) 

where  dB/dn  is  the  angle  subtended  per  fringe  for  the  wave-length  X,  the 
grating  space  D,  the  thickness  of  air-space  e,  at  an  angle  of  incidence  i  and 
of  diffraction  6,  in  air. 

These  three  sets  need  not  be  in  focus  at  once.  Equations  4  and  6  are 
usually  easily  put  in  focus  together.  Thus  in  the  above  case  roughly, 

«  =  5o°24'  #  =  24°  44'  ,*/£>=. 352 

whence  for  X  =  6oXio~6  cm.  and  e=i  cm., 

dO'  _  o6  ,  d6"  ^  df)  =     _g, 

dn  dn  dn 

In  case  of  the  set  of  equations  4,  5,  6,  therefore,  there  should  be  about 
6  to  7  small  fringes  to  one  large  fringe.  This  is  about  the  order  of  values 
usually  observed.  When  e  is  small  the  change  of  wave-length  with  dd  must 
be  considered.  To  obtain  a  given  ratio  k  of  small  and  large  fringe  diver- 
gences, one  may  write  for  the  cases  4  and  6,  for  instance, 

,  _  i  —  cos  (i  —  6) 

''  *  COS  ~(*  -  0)  +  COS~  (t +0)"  (  7  ' 

Equation  7  is  not  easily  treated.  If,  however,  8  is  computed  in  terms  of 
i  and  expressed  graphically,  k  may  then  also  be  expressed  in  terms  of  i;  and 
thus  the  angle  of  incidence  i  for  any  ratio  of  size  of  fringes,  k,  in  question, 
may  be  roughly  adjusted.  Table  7  shows  these  results. 


IN    RELATION    TO    INTERFEROMETRY. 
TABLE  7. — Ratio  of  large  and  small  fringes. 


33 


i 

0        i/k 

\ 

i/k' 

i 

9 

i/k 

i/k' 

0° 

-20°   l'      16.4 

i6-5 

SO° 

24°  27' 

6.0 

7.0 

IO 

—  io  17     15.6 

16.6 

60 

3°  56 

3-4 

4-4 

20 

-  o  34     14.8 

15-8 

70 

36   o 

1.6 

2.6 

3° 

+831     12.3 

13-3 

80 

39  15 

.6 

i-5 

40 

16  54      9.1 

10.  2 

90 

40  24 

.0 

I  .0 

Thus  it  appears  that  at  ^  =  37°  about  there  should  be  ten  small  fringes 
to  one  large  fringe.  In  a  general  way,  moreover,  the  ratio  of  small  to  large 
fringes  gives  an  estimate  of  the  value  of  i. 

Similarly  equations  5  and  6  give 


I~cos   *- 


-  _ 

2  -  (cos  (i  -  6}  -  cos  (*+0)) 

from  which  the  data  also  given  in  table  i  ,  follow. 

Here  there  will  be  ten  small  to  one  large  fringe  when  i  is  40.5°  roughly. 

The  preceding  equations  4,  5,  and  6  are  essentially  approximate,  inas- 
much as  the  rates  are  taken  for  the  finite  quantities.  If  we  return  to  the 
fundamental  equations 

2€  COS  I  =  tt/  (8) 

2ecosO  =  n'/>  (9) 

sin  0  —  sin  i  =  XjD  (io) 

for  6  is  greater  than  i,  where  n  and  n'  are  unequal  distributive  whole  num- 
bers; and  if  we  put  e/D  =  a,  we  find 


. 
tan  2  =  —      a 


4<3« 


+  a 


(II) 


which  are  free  fromX,  whereas  sin  i  and  sin  6  essentially  contain X.    As  the 
angular  width  of  a  fringe  is 

M* 

T~  dn 
dn 

those  corresponding  to  equations  4  and  5  may  therefore  be  expressed  as 

-dn 


0,  =  2a   I  - 

Jn  "'     »'- 

02=2(1  n(n- 

*/n 


4(in  tan  i  —  4a2 


(12) 


_ 
2a  tan  i) 


for  a  given  space  e  in  a  =  b/D  and  a  given  angle  of  incidence  i  in  tan  *. 
These  integrations  are  easily  made,  but  the  results  are  too  diffuse  to  be 


34  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

worth  discussing  here.    Equation  12,  moreover,  is  a  restatement  of  the  fact 
that  in  these  interferences  n\  is  constant  throughout  the  spectrum. 

Equations  8,  9,  and  10,  however,  admit  of  the  graphic  treatment  of  the 
problem.  For  if  we  put  A=nD/ze,  they  may  be  written  according  as 
9>i  or  6<i 

sin0l-±—   -+sin*  (14) 

/i 


<«S> 


t+A. 


where  the  distributive  number  n  in  A  takes  the  values  of  the  successive 
whole  numbers  for  a  dark  band  in  the  respective  spectra  corresponding  to 
equations  14  and  15.    The  coincidence  of  dark  bands  then  determines  the 
position  of  the  coarse  fringes. 
If  i-  50°  24' 

2£=.icm.        D=  .000,169  cm.        #  =  24°  44' 
then 

A  =  .0017  •  M,  nearly 

With  these  data  table  8  was  computed.  Similar  results  might  be  computed 
for  6<i,  but  I  have  abandoned  it  because,  as  is  now  evident,  the  practical 
demands  are  sufficiently  met  by  the  method  of  paragraph  28. 

TABLE  8.  —  Showing  tf,  and  n,  "2  and  n'.     H  observed  about  75°.     2^  =  i  cm.,  *  =  5o.4°, 
0  >  i.     tie  corresponds  to  the  mean  value  8. 


n 

". 

10'fo, 

», 

I 
n' 

'. 

io*8ea 

i 

1650 

86. 

1° 

76 

84.2 

°     30° 

83.40 

19 

82.5° 

1700 

82. 

3 

46 

81.2 

400 

Sl.S 

18 

80.5 

175° 

80. 

0 

36 

79.1 

500 

79.6 

i7 

78.8 

1800 

78. 

2 

3° 

77.6 

600 

77-9 

16 

77.1 

1840 

77- 

0 

23 

76.7 

700 

76.2 

15 

75-5 

1870 

76. 

3 

30 

75-8 

800 

74-7 

14 

74-0 

1900 

75- 

4 

22 

75-0 

900 

73-3 

12 

72.7 

IQ40 

74- 

5 

20 

74-2 

IOOO 

72.1 

1970 

73- 

9 

2O 

73-6 

20OO 

64  .  o 

2OOO 

73- 

3 

18 

73-2 

2O2O 

73- 

0 

20 

72.8 

.  .  . 

2O40 

72. 

6 

15 

72.5 

.  .  . 

1 

29.  Test  made  by  magneto-striction. — The  very  small  elongation  pro- 
duced when  iron  is  magnetized  offers  an  excellent  test  of  the  above  appa- 
ratus. The  attached  water-cooled  helix  has  already  been  described.  The 
rod  of  Swedish  soft  iron  was  quite  within  the  helix,  which  surrounded  it 
closely  without  contact,  the  rod  being  prolonged  by  light  copper  tubes 
soldered  to  its  ends. 


IN    RELATION    TO    INTERFEROMETRY. 


35 


A  large  number  of  experiments  were  made  for  trial,  brief  examples  of 
which  are  given  in  the  following  table  and  charts,  where  if  the  current  is 
in  amperes  the  magnetic  field 

H=io8i  gauss 

and  the  elongation  5e=io$Xio~6  cm.  per  large  fringe.  Thus  if  n  is  the 
number  of.  fringes  and  ^  =  28  cm.  the  effective  length  of  the  iron  rod,  the 
absolute  elongation  is  da=  105^X1  o"6  cm.  and  the  elongation  per  unit  of 
length, 


or  if  small  fringes  N  are  taken, 

dp  =.  54^X10-" 

Elongations  from  lo"6  cm.  to  io~4  cm.  were  measured  without  difficulty, 
though  such  measurement,  without  micrometer  attachment,  is  essentially 
an  estimate.  Count  was  made  of  the  number  of  small  fringes  between  the 
large  fringes. 

TABLE  9.  —  Magnetostriction  elongations.  Coil*  (separated  by  thin  sheet  of  flow- 
ing water  from  rod)  length  37  cm.,  3200  turns,  H  =  io8t  gauss  (*'  in  amperes). 
Swedish  iron  rod,  diam.,  2r  =  .64  cm.,  length  /  =  30.6  cm.,  free  length  28  cm. 
Elongations,  10*^  =  105  cm.,  io6&?i=33  cm.,  io6<te2  =  47  cm.  Seven  small  fringes 
to  one  large  fringe. 


H 

Fringes. 

^Xio" 

H   \  Fringes. 

«5£Xio8 

gauss 

small 

gauss   small 

13 

.0 

0 

98    6.0 

320 

22 

i.  5 

79 

56  i  7-° 

37° 

36 

4.0 

2IO 

36    5-o 

310 

44 

6.0 

320 

23    2.0 

I  10 

62 

7.0 

37° 

9     -o 

0 

62 

6.5 

350 

-  9     -5 

26 

44 

5-5 

300 

-23    3.0 

160 

36 

5-° 

270 

-36  j   5-5 

300 

13 

.0 

0 

—  56    6.0 

320 

-13 

•  5 

26 

-99    5-0 

270 

—  22 

3-5 

190 

+  99    5-o 

270 

-62 

7.0 

37° 

69  j  6.0 

320 

+  62 

7.0 

37° 

56    6.0 

320 

'3 

•5 

26 

19  i   1.2 

64 

6 

.0 

0 

ii   .    .0 

0 

I"  !  -5 

26 

*  A  field  of  150  gauss  should  more  than  practically  saturate  iron. 

The  data  partake  of  the  usual  character  of  magnetic  phenomena.  There 
is  marked  hysteresis  and  irregularity  due  to  residual  magnetization.  Table 
9  has  been  inserted  merely  as  an  example  of  the  results,  which  are  otherwise 
given  graphically.  The  maximum  elongation  corresponded  to  about  7 
small  fringes. 

The  curves  bring  out  the  following  results  consistently: 
(i)  The  elongation  apparently  begins  abruptly  in  small  fields  of  5  or  10 
gauss,  see  fig.  12,  etc.,  at  b. 


36 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


(2)  In  small  fields,  fig.  14,  A,  there  is  nevertheless  a  tendency  of  a  very 
slow  continual  increase;  but  whether  this  is  due  to  inevitable  percussion 
during  magnetization  can  not  be  stated.    The  case  may  be  one  of  true  mag- 
netic elasticity. 

(3)  The  maximum  elongation  is  very  rapidly  reached  in  a  field  of  50  to 
100  gauss.    After  this  the  elongation  now  due  to  strong  fields  decreases. 
In_the  case  of  strong  fields,  however,  the  rod  as  a  whole  is  probably  dis- 


SO     -60     -40      20 


20      40       60      80 


FIG.  12. — Charts  showing  the  elongation  of  soft  iron  in  millionths  for 
different  fields,  positive  and  negative,  in  gauss.  Field  applied  and 
removed  for  each  observation. 


Wari 


M 


80    -60     40      £0       0       20      40      60      80 

FIG.  13. — Chart  showing  the  elongation  of  soft  iron  in  millionths  for 
different  fields,  positive  and  negative,  in  gauss.  Field  applied  and 
removed  for  observation. 


IN    RELATION    TO    INTERFEROMETRY. 


37 


placed  or  warped.  The  results  are  less  and  less  trustworthy  and  not 
suitable  for  exploration  by  the  optic  method  in  question  (figs.  12,  B,  and 
14,  B  and  C). 

(4)  If  the  field  is  reversed  the  elongation  is  similar,  but  symmetrical 
•with  respect  to  a  small  positive  field  if  the  immediately  preceding  magneti- 
zation was  positive,  and  symmetrical  with  respect  to  a  small  negative  if 
the  preceding  magnetization  was  negative.  See  figs.  12,  A,  B,  13  (hystere- 
sis). Thus  in  fig.  12,  A  and  B,  where  a  field  of  13  gauss  at  b  after  a  positive 
magnetization  produces  no  elongation  whatever,  the  reversal  of  the  field 
to  — 13  gauss  produces  a  very  marked  elongation  of  nearly  3  X  io~7  which 


0      10       20     SO     40       50     60 


FIG.  14. — Charts  showing  the  elongation  of  soft  iron  in  millionths  for 
different  fields,  positive  and  negative,  in  gauss.  Field  applied  and 
removed  for  each  observation  except  in  fig.  D. 


2091 


38  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

may  be  repeated  indefinitely.  Hence  there  seems  to  be  a  definite  magnetic 
elasticity  with  fields  of  less  than  5  to  10  gauss.  If  this  is  exceeded  the  mag- 
netic molecules  are  permanently  set,  even  in  soft  iron. 

(5)  If  the  field  varies  continuously  (liquid  resistances)  the  elongations 
are  much  smaller  in  given  fields  (fig.  14,  D). 

The  whole  subject,  which  is  here  touched  incidentally,  is  well  summar- 
ized in  Winkelmann's  Handbuch,  vol.  5,  p.  307  et  seq.,  1908,  by  Professor 
Auerbach.  The  above  results  agree  closely  with  the  elaborate  experiments 
of  Nagaoka  and  Honda,  *  who  used  a  special  type  of  contact  lever.  Recently 
Mr.  Dorsey  t  has  made  a  similar  investigation,  using  an  ingenious  method 
of  his  own.  The  present  method  would  not  have  been  feasible  but  for  the 
occurrence  of  large  and  small  fringes  in  the  field  of  the  telescope. 

30.  Summary.— The  present  chapter  has  shown  the  ease  with  which  the 
above  phenomena  may  be  adapted  to  the  measurement  of  small  displace- 
ments, has  instanced  their  particular  usefulness,  inasmuch  as  fine  and 
coarse  adjustments  are  both  available  (recommending  their  use  in  the 
investigation  of  phenomena  of  sudden  occurrence  as  in  magneto-striction, 
etc.),  and  has  given  suggestions  of  practical  forms  of  apparatus  in  addition 
to  the  usual  spectrometric  form. 

*  Nagaoka  and  Honda:  Phil.  Mag.  (5),  xxxvn,  p.  131,  1894. 
f  Dorsey:  Phys.  Rev.,  xxx,  p.  698,  1910. 


CHAPTER  IV. 


THE  USE  OF  THE  GRATING  IN  INTERFEROMETRY ;  EXPERIMENTAL 
RESULTS. 

31.  Introductory.— In  chapters  II  and  III  a  method*  was  described  of 
bringing  reflected-diffracted  and  diffracted-reflected  rays  to  interference, 
producing  a  series  of  phenomena  which,  in  addition  to  their  great  beauty, 
promise  to  be  useful.  In  fact,  the  interferometer  so  constructed  needs  but 
ordinary  plate  glass  and  replica  gratings.  It  gives  fringes  rigorously 
straight,  and  their  distance  apart  and  inclination  are  thus  measurable  by 
ocular  micrometry.  Lengths  and  small  angles  are  thus  subject  to  micro- 
metric  measurement.  Finally,  the  interferences  are  very  easily  produced 
and  strong  with  white  light,  while  the  spectrum  line  used  may  be  kept  in 
the  field. 


3 


FIG.  15. — Diagram  of  grating  interference  adjustments  adapted  to 
Michelson's  apparatus.  L,  source  of  white  light  (parallel  rays),  gg' 
grating,  M  and  N  opaque  mirrors. 

The  same  method  is  available  as  an  adjunct  to  either  Jamin's  or  Michel- 
son's  interferometers,  except  that  here  the  transmitted-reflected-diffracted 
and  reflected- transmitted-diffracted  rays  are  brought  to  interfere.  To 
take  the  example  of  the  Michelson  type,  stripped  of  unnecessary  details, 
let  gGg'  in  fig.  15  be  the  grating  or  ruled  surface,  n  its  normal,  L  the  source 
of  white  light,  M  and  N  the  mirrors,  and  E  the  eye.  In  the  usual  way  the 
rays  from  L  interfere  at  E. 

*Science,  xxxii.p.Qa,  1910;  cf.  Phil.  Mag.,  July  1910,  and  chapters  II  and  III  above. 


40  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

Now  replace  L  by  a  slit  and  collimator,  £  by  a  telescope  focussed  for 
parallel  rays.  The  eye  at  E  now  sees  a  sharp  line  of  light.  At  D  and  D', 
however,  there  must  be  two  diffraction  spectra  coinciding  in  all  their  parts 
and  hence  interfering  rhythmically,  if  all  adjustments  are  sufficiently  per- 
fected. The  other  two  diffractions  within  MGg'  and  EGg  are  often  lost  at 
an  incidence  of  45°. 

The  attempt  to  produce  these  interferences  at  D,  D'  with  replica  gratings 
is  liable  to  result  in  failure;  for  while  the  transmitted  system  NGD  shows 
brilliant  spectra,  the  reflected  system  MGE  is  dull  and  hazy.  Both  spectra 
are  clearly  in  evidence  and  may  be  brought  to  overlap.  The  film,  however, 
does  not  reflect  in  a  degree  adequate  to  the  transmission.  Attempts  made 
to  realize  the  conditions  of  interference  eventually,  however,  resulted  in 
complete  success  (chapter  V). 

What  is  strikingly  feasible  at  once,  with  ordinary  plate  glass  and  a  non- 
silvered  grating,  *  is  the  production  of  interferences  between  pairs  of  dif- 
fracted spectra,  D'  and  D,  for  instance,  if  returned  by  equidistant  mirrors 
M  and  AT  to  a  telescope  in  the  line  D.  Both  of  these  spectra  are  very  bril- 
liant and  not  very  unequally  so  and  the  coincidence  of  spectrum  lines  both 
horizontally  and  vertically  brings  out  the  phenomenon.  This,  for  each  of 
the  cases  specified,  is  of  the  ring  type  and  not  of  the  line  type  heretofore 
discussed;  but  it  occupies  the  whole  field  of  the  spectrum  from  red  to 
violet.  One  obtains  brilliant  large  confocal  ellipses  with  horizontal  and 
vertical  symmetry,  and  the  spectrum  lines,  simultaneously  in  focus,  may 
serve  either  as  major  or  minor  axes.  The  interferometer  motion  is  twofold 
in  character,  consisting  of  radial  motion,  combined  with  a  drift  of  the  figure 
as  a  whole,  in  a  horizontal  direction.  Naturally  a  fine  slit  is  of  advantage, 
but  the  experiment  succeeds  with  quite  a  wide  slit,  especially  in  the  red, 
much  after  spectrum  lines  vanish.  Similarly  the  regular  reflection  from 
M  and  A'  (mirrors)  will  produce  these  phenomena  along  GD. 

Such  ellipses  (I  shall  so  call  them,  though  they  are  probably  ovals)  as 
have  their  centers  in  the  field,  are  clearly  due  to  reflection  from  the  same 
surface,  as  shown  in  fig.  1 5 ;  curved  lines  or  ellipses  with  remote  centers  are 
due  to  simultaneous  reflections  of  the  component  rays  from  the  opposite 
faces  of  the  grating,  since  the  angle  of  the  wedge  of  glass  can  not  be  excluded. 
All  owe  their  vertical  and  horizontal  symmetry  to  the  vertical  slit  and 
horizontal  spectrum.  The  ellipses  are  identically  present  in  the  successive 
orders  of  spectra  at  once. 

These  elliptical  fringes  thus  embody  with  the  preceding  linear  set  the 
common  property  of  being  duplex  in  character;  only  here  the  motion  of  dark 
rings  to  or  from  the  centers  of  the  ellipse,  as  a  fine  adjustment,  is  associated 
with  a  displacement  of  the  fringes  bodily  through  the  spectrum  (coarse 
adjustment). 


*  My  thanks  are  due  to  Prof.  J.  S.  Ames,  of  Johns  Hopkins  University,  who  was 
good  enough  to  lend  me  this  grating. 


IN    RELATION    TO    INTERFEROMETRY.  41 

This  displacement  may  be  from  red  to  violet  or  from  violet  to  red,  as  the 
virtual  air-space  increases  in  thickness,  depending  upon  the  adjustment, 
as  will  presently  appear.  In  other  words,  for  a  given  small  interferometer 
motion  of  mirror,  there  is  in  general  less  displacement  of  fringes  bodily 
•than  radial  motion  of  fringes  to  or  from  the  center.  Slight  deflection  of  the 
grating,  gg  ,  is  chiefly  accompanied  by  radial  motion  of  the  fringes.*  The 
effect  thus,  produced  is  to  give  sharpness  to  the  interference  pattern,  which 
soon  vanishes  for  approximate  adjustments  of  spectra. 

Similarly  the  micrometer  screw  produces  a  rapid  passage  of  the  pattern 
through  its  maximum  size,  a  play  of  the  screw  of  o.  i  cm.  being  sufficient  to 
pass  from  fringes  of  one  extreme  of  small  size,  through  the  maximum  size 
to  the  final  extreme.  One  may  note  that  in  chapters  II  and  III  the  admis- 
sible play  for  the  coarse  fringes  was  about  i  cm.,  ten  times  greater.  On  the 
other  hand  there  are  many  regions  of  interference;  in  fact, different  groups 
of  interferences  may  sometimes  be  in  the  field  at  once,  or  more  usually 
corresponding  to  slightly  different  angles  between  the  mirrors  and  grating. 

All  admissible  angles,  provided  that  symmetry  is  maintained  (the  virtual 
plane  of  the  grating  bisecting  the  angle  of  the  mirrors) ,  bring  out  the  phe- 
nomenon. This  points  to  the  fact  that  the  earlier  phenomenon  referred 
to  (I.e.),  in  which  the  center  or  normal  ray  was  virtually  at  an  infinite  dis- 
tance, is  now  produced  near  an  accessible  center,  even  if  this  is  not  actually 
in  the  field.  However,  in  the  earlier  case,  when  the  angle  of  incidence  is 
zero,  curved  lines  may  also  be  obtainable.  Such  cases  will  be  given  in  the 
next  chapter. 

Experiments  with  a  silvered  grating  showed  no  advantage,  whether  the 
transparent  film  covered  the  grating  or  the  plane  face  of  the  glass.  In 
fact,  in  case  of  good  adjustment  the  phenomenon  is  so  strong  as  to  need  no 
accessory  treatment;  this  is  in  fact  one  of  its  advantages.  A  great  diffi- 
culty in  adjustment  is  the  occurrence  of  stationary  fringes  due  to  the  rear 
face  of  the  grating.  These  may  even  wipe  out  the  spectrum  lines;  but 
usually  they  lie  at  a  finite  focus  and  are  not  seen  with  a  sharp  spectrum. 
Fortunately  many  positions  are  available  for  obtaining  the  ellipses,  so  that 
a  satisfactory  one  is  easily  found.  Unless  all  overlapping  spectra  show 
sharp  lines  the  adjustment  is  very  tedious. f  The  amount  of  grating  space 
used  is  less  than  i  sq.  cm.,  though  of  course  a  larger  size  is  convenient  for 
experiment. 

32.  Special  properties. — The  motion  of  the  ellipses  bodily  across  the 
spectrum,  corresponding  to  the  increased  or  decreased  virtual  air-space 
(micrometer  screw),  shows  that  whereas  the  vertical  dimension  or  axes 
(direction  of  fixed  color)  do  not  appreciably  change,!  the  horizontal  axes 

*  This  changes  the  effective  thickness  of  the  grating. 

t  These  difficulties  are  largely  removed  by  the  method  of  coincident  white  slit 
images  given  in  chapter  V. 

J  Probably  referable  to  increased  refraction  and  decreased  diffraction  from  red 
to  violet. 


42  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

grow  rapidly  smaller;  i.e.,  the  ellipse  is  more  eccentric  from  red  to  violet. 
With  the  grating  used,  however  (about  2800  lines  per  centimeter),  the 
major  axis  remained  vertical,  i.e.,  the  circular  form  was  not  reached 
throughout  the  first  order  even  in  extreme  red. 

It  follows  from  this  that  in  the  second  order  the  major  axis  of  the  ellipses 
for  2800  lines  to  the  centimeter  will  probably  be  horizontal,  and  this  was 
found  to  be  strikingly  true.  The  figure,  moreover,  is  necessarily  coarser  and 
the  lights  and  shadows  in  greater  contrast.  In  the  third  order  the  ellipses 
are  drawn  out  horizontally  to  a  correspondingly  greater  degree.  Thus  it  is 
clear  that  with  a  more  dispersive  grating  the  circular  or  even  the  hori- 
zontally elongated  form  must  occur  in  the  first  order.  Indeed  with  6000 
lines  to  the  centimeter,  ellipses  occur  in  the  red  having  horizontal  major 
axis  while  the  figure  in  the  green  is  circular.  Spectrum  lines  are  very  dis- 
tinct, particularly  in  the  second  order.  In  all  orders  of  spectra  the  centers 
of  the  ellipses  are  simultaneously  on  the  same  spectrum  line  and  their 
vertical  dimensions  are  about  the  same.  Hence  the  spectrum  lines  may  be 
used  instead  of  cross-hairs,  as  they  are  fixed  landmarks  among  the  moving 
ellipses. 

With  replica  gratings  the  center  of  the  ellipses  is  usually  remote,  i.e., 
reflection  does  not  easily  take  place  at  the  free  grating  surface.  Apart 
from  this  the  curved  lines  are  good  and  strong.  Samples  must  be  tried  out, 
in  which  the  lines  are  as  clear  as  possible  in  all  the  overlapping  spectra.  Six 
gratings  of  this  kind  were  examined  with  about  the  same  results.  The 
centers  of  the  ellipses  could  only  rarely  be  brought  into  the  field.  Centers 
may,  however,  be  obtained  for  gratings  cemented  under  pressure  between 
unequally  thick  plates  of  glass. 

It  has  been  stated  that  considerable  width  of  slit  is  admissible.  Moreover, 
the  focal  plane  of  the  collimator  (convergent  or  divergent  light)  or  the  focus 
of  the  telescope  makes  little  difference,  though  the  condition  of  parallel 
rays  is  naturally  preferable.  An  eye  focussed  for  infinity  sees  the  fringes 
very  well  without  the  telescope  and  they  may  be  also  caught  on  a  screen; 
but  coming  through  a  slit  they  are  liable  to  be  dark  and  the  advantage  of 
the  telescope  is  obvious.  The  second  order  is  particularly  accessible  for 
naked-eye  observation,  and  the  light  and  black  appearance  is  accentuated, 
but  they  are  liable  to  be  irregular.  Inclining  the  collimator  moves  the 
spectrum  across  the  fringes  vertically,  while  inclination  of  the  telescope 
moves  both  equally.  In  this  way  the  centers  may  often  be  found.  With 
strong  tipping,  however,  the  figure  becomes  distorted  or  open  above  and 
below,  as  would  be  expected.  If  the  light  is  intensified  for  projection  or 
naked -eye  work,  there  is  also  distortion. 

It  is  not  necessary  (and  apparently  of  little  advantage)  to  have  the 
reflections  from  the  mirrors  M  and  N  occur  at  normal  incidence.  In  fact 
the  patch  of  white  light  on  the  grating  surface  and  the  return  patch  of 
spectrum  may  be  over  an  inch  apart.  Inasmuch  as  the  spectra  are  rigo- 
rously coincident,  it  follows  that  (apart  from,  the  refraction  of  the  thick 


IN    RELATION    TO    INTERFEROMETRY.  43 

plate  glass)  the  grating  must  be  symmetrical  with  respect  to  the  mirrors; 
i.e.,  intersect  the  angle  between  them.  Very  little  difference  of  size  or  shape 
is  observable  between  extremes  of  such  adjustment.  Thus  I  rotated  the 
grating  about  10°  (larger  angles  being  unavailable)  without  appreciable 
effect,  after  one  of  the  mirrors  had  been  correspondingly  adjusted. 

On  examination  of  the  effective  air-distances  of  the  mirrors  M  and  N 
from  the  grating,  it  is  found  that  three  positions  are  favorable  to  inter- 
ference; in  the  first  (case  A),  M  is  farther  away  than  N  from  the  corre- 
sponding faces  of  the  glass  plate;  in  the  second  (case  B)  they  are  about 
equidistant;  finally,  in  the  third  (case  C),  N  is  further  away,  symmetrically 
with  the  first  adjustment.  For  each  case  the  admissible  play  of  mirror 
(micrometer  screw)  does  not  much  exceed  i  cm.  for  ordinary  magnification. 
The  second  or  equidistant  adjustment  brings  out  the  lines  or  ellipses  with 
distant  centers,  and  there  are  usually  three  types  easily  found  (after  one 
has  appeared),  by  slightly  inclining  either  mirror  around  a  horizontal 
axis  by  a  tangent  screw.  Thus  (see  fig.  16)  fine  lines,  say  at  135°  to  the 


FIG.   1 6. — Interference  patterns  in  case  of  self-compensation,  B. 

horizontal,  coarse  nearly  circular  lines,  concave  downward  or  upward  as 
the  case  may  be,  and  finally  broad  lines  say  45°  to  the  axis,  may  appear 
in  succession.  If  the  grating  is  rotated  180°  around  its  normal,  convexity 
upward  (fig.  16,  A)  changesto  concavity  upward  (fig.  16,  B),  showing  the 
wedge  angle  of  the  glass  plate  to  be  effective.  The  micrometer-screw  will 
pass  any  of  these  forms  through  a  succession  of  inclinations  corresponding 
to  the  eccentric  intersection  of  a  group  of  concentric  circles  by  a  straight 
line.  This  is  also  shown  in  fig.  1 6  at  A  and  B,  as  observed  for  a  fixed  air- 
space and  at  C,  on  moving  the  micrometer  in  the  A  and  B  cases.  These 
figures  cover  the  coincident  spectrum  fields  only.  A  solitary  part  of  the  field 
shows  no  interferences.  For  adjustment  it  is  therefore  necessary  to  cut  off 
the  spectra  sharply  above  and  below,  by  limiting  the  length  of  slit.  Of 
the  three  or  four  spectra  present  the  coincidence  may  then  be  secured  with 
least  difficulty.  With  each  such  coincidence  there  goes  a  definite  position 
of  the  micrometer  screw  (interferometer),  and  this  is  the  initial  difficulty 
4  - 


44  THE    PRODUCTION    OF   ELLIPTIC    INTERFERENCES 

of  adjustment;  i.e.,  to  coordinate  the  various  spectrum  coincidences  with 
the  three  screw  readings. 

The  non-equidistant  case  of  mirrors  and  grating  correspond  to  the  ring 
types  with  the  centers  in  the  field.  Hence  both  reflections  take  place  from 
the  same  face  of  the  grating.  When  the  distance  GM  is  shorter,  G'N  longer, 
I  have  noticed  two  or  even  three  ellipses  successively  in  the  field,  obtained 
by  slowly  inclining  the  mirror  N  about  the  vertical  (tangent  screw),  with 
their  centers  in  turn  in  the  yellow-green,  the  blue,  and  the  green  of  the 
spectrum.  They  do  not  occur  together.  On  turning  the  micrometer  screw 
clockwise  they  move  from  red  to  violet,  and  vice  versa. 

When  the  distance  GM  is  longer  and  G'N  shorter,  rings  also  occur;  but 
only  a  single  set  was  found.  The  spectrum  lines  not  being  clear,  finding 
them  was  a  matter  of  discrimination.  When  the  micrometer  screw  was 
turned  clockwise,  however,  they  marched  from  violet  to  red,  i.e.,  in  opposite 
direction  to  the  preceding.  The  drift  of  ellipses  corresponds  with  the 
direction  in  which  the  vjolet  fringes  move  horizontally. 

It  is  at  first  an  astonishing  result  that  when  the  grating  is  reversed 
(front  face  put  rearward,  leaving  the  air  space  unchanged)  the  ring  type 
is  left  in  the  field ;  though  the  solitary  ring  type  changes  to  the  multiple 
type.  This  is  true  for  the  case  specified  as  A  or  case  C.  It  is  also  true  for 
the  eccentric  type,  case  B,  where  line  types,  single  or  multiple,  reappear  at 
slightly  different  mirror  angles.  Briefly,  rotation  has  no  effect  on  the  march 
of  ellipses  through  the  spectrum;  but  in  case  B  it  changes  convex  lines 
downward  to  concave  lines,  and  vice  versa.  Nor  has  reversal  of  grating  any 
effect  on  the  march.  The  position  of  the  grating  with  respect  to  the  mirrors 
(three  places  being  available)  alone  determines  this  result,  certainly  in  the 
extreme  cases  A  and  C,  and  probably  in  the  intermediate  case  B. 

The  use  of  a  compensator  in  either  of  the  component  rays  is  always 
accompanied  by  the  two  effects  in  question;  i.e.,  there  is  both  relatively 
large  radial  motion  of  the  fringes  and  relatively  small  displacement,  within 
the  field  of  the  telescope.  One  may  therefore  be  evaluated  in  terms  of  the 
other,  the  two  having  the  stated  relation  of  coarse  and  fine  adjustments. 
In  case  B  the  fringes  will  rotate,  expand,  or  contract. 

33.  Elementary  theory.— The  endeavor  must  now  be  made  to  explain 
these  results  more  in  detail.  For  this  purpose  fig.  1 7  may  be  consulted. 

In  the  grating  used  at  an  incidence  of  about  45°  the  angle  of  diffraction 
in  the  first  order  was  about  32°  39'  for  the  A  sodium  line,  there  being 
about  2847  lines  to  the  centimeter,  corresponding  to  a  grating  space  of 
D  = .  0003  5 1 2  cm .  Thus  \/D  =  0.1677.  The  index  of  refraction  was  assumed 
to  be  1.53  for  the  same  line. 

The  diagrams  are  drawn  for  an  angle  of  incidence  of  45°  and  for  normal 
reflection  from  the  mirrors  M  and  N.  In  case  of  the  grating  given,  the 
three  positions  of  the  grating  at  which  interferences  occur  were  about  0.6 
cm.  apart  and  are  marked  124,  100,  and  76  scale  parts,  on  the  micrometer 


IN    RELATION    TO    INTERFEROMETRY. 


45 


screw  normal  to  the  grating.  In  other  words  it  was  convenient  to  move  the 
grating  at  the  center  of  the  spectrometer,  rather  than  the  mirrors  M  and  N 
adjustably  attached  near  the  edge  of  the  plate  graduated  in  degrees. 

In  group  A  at  124  the  short  air-path  is  to  M,  the  long  path  to  N;  in  group 
C  at  76  the  reverse  is  true.  In  group  B  at  100,  the  system  is  self-compen- 
sating and  the  air-paths  about  equal. 

In  all  cases  the  angle  of  diffraction,  6,  is  less  than  the  angle  of  incidence,  i. 
There  are  of  course  corresponding  cases,  6>i,  which  have  not  been  drawn, 
because  they  merely  duplicate  the  cases  given,  at  a  different  angle.  It  is 
assumed  that  after  more  than  one  direct  reflection  or  one  diffraction  the 
interferences  are  no  longer  observable. 


FIG.   17. — Chart  of  the  three  groups  of  interferences.  A,  B,  C.    Mirrors  M  and  N 
return  the  spectrum.     Grating  face  in  position  shown  at  g. 

The  diagrams  124  and  76  show  that  the  two  reflections  of  the  component 
rays  at  the  grating  take  place  at  the  same  surface;  hence  the  occurrence  of 
centered  figures  or  rings.  On  the  contrary  the  reflections  in  diagram  100 
take  place  at  the  two  different  faces  of  the  grating  respectively;  hence 
the  angle  of  the  grating  is  included  and  liable  to  produce  eccentric  ring 
systems.  The  center  may  be  so  far  off  that  the  dark  lines  are  nearly  straight, 
but  they  change  their  inclination  as  the  vertical  projection  of  the  center 
moves  horizontally  through  the  field. 

Some  of  these  cases  may  coalesce  in  practice  or  they  may  destroy  each 
other  more  or  less.  I  have  taken  a  single  incident  ray  from  which  may 
come  two  parallel  emergent  rays,  which  are  brought  to  interfere  by  the 
telescope.  It  would  have  been  just  as  convenient  to  have  taken  the  two 


46  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

corresponding  incident  rays  which  interfere  in  a  single  emergent  ray.  From 
the  position  of  the  mirrors  it  is  clear  that  the  regularly  refracted  rays  are 
not  returned.  Only  rays  first  diffracted  at  the  grating  (where  they  may  also 
be  reflected)  are  returned  by  the  mirrors. 

As  a  whole  we  may  distinguish  two  typical  cases,  those  in  which  both 
component  rays  are  diffracted  as  in  No.  i  or  refracted  as  in  No.  2 ;  and  those 
in  which  one  component  ray  is  refracted  and  the  other  diffracted.  If  *  and 
8'  are  the  angles  of  incidence  and  diffraction  in  air  and  r  and  0  the  corre- 
sponding angles  of  refraction  and  diffraction  in  glass,  the  glass  path  differ- 
ences, Jx,  in  the  important  cases  are  as  follows : 

No.  i.  Jx  =  2ne/cos  6=2.2  cm.  No.  7.  J#  =  Zero. 

3.  =2/u£/cos  6=2.2  cm.  8.  =Zero. 

4.  =Zero.  9.  =  —  2/ztf/cos  6  =  2.2  cm. 
6.  =Zero.  10.  =  —  zjue/cos  6  =  2.2  cm. 

Here  M  is  the  index  of  refraction  and  e  the  thickness  of  the  glass  plate  of 
the  grating,  and  excess  of  path  for  the  M  ray  is  reckoned  positive.  These 
paths  must  be  compensated  by  corresponding  decrements  and  increments 
respectively  of  the  air  paths  GM  and  GN.  Ordinarily  these  path  differences 
in  glass  being  fixed  for  given  angles  tf  would  fall  away;  but  they  vary  essen- 
tially with  color  and  hence  the  degree  of  compensation  is  never  the  same 
for  all  colors. 

Furthermore,  although  the  wave-fronts  of  the  two  rays  are  the  same  on 
emergence,  this  does  not  imply  coincidence  of  phase  even  in  such  cases  as 
Nos.  i  and  2,  for  instance;  the  absolute  lengths  of  paths  in  glass  are  quite 
different,  although  their  differences  are  the  same.  Consequently  the  cases 
i  and  2  would  again  interfere  if  superimposed,  one  case  being  first  diffracted 
and  the  other  first  refracted. 

Thus  it  is  not  surprising  that  so  many  cases  were  identified.  It  is  also 
apparent  that  the  air  compensations  are  very  different  and  hence  identifi- 
cation is  facilitated.  Finally,  since  a  vertical  slit  and  collimator  are  used, 
the  section  of  a  beam  of  light  passing  through  the  grating  by  a  horizontal 
plane  consists  of  parallel  rays;  the  section  by  a  vertical  plane,  however,  is 
convergent. 

It  is  interesting  to  find  the  numerical  data  for  the  above  equations, 
assuming  that 

«  =  45°        0'  =  32° 39'         *=.68cm.        M=  i. 5 3  (estimated)       \/D=.i6jj 

for  the  sodium  line  of  the  spectrum.  The  results  are  given  with  the  equa- 
tions. Their  value  is  about  2.2  cm.,  which  is  equivalent  to  a  displacement 
of  mirror  actually  found. 

It  follows,  moreover,  that  the  center  of  the  ring  system,  order  n  =  o, 
must  move  from  red  to  violet  or  the  reverse,  inasmuch  as  the  compensation 
takes  place  successively,  at  each  color,  in  the  same  way.  Although  the 
equations  hold  only  for  the  center,  and  the  symmetrically  oblique  rays 


IN    RELATION    TO    INTERFEROMETRY. 


47 


belonging  to  the  rings  have  not  been  given  consideration,  an  approximate 
computation  of  the  motion  of  the  ring  centers  may  nevertheless  be 
attempted. 

From  the  equations  qualitative  interpretations  of  the  above  and  the  fol- 
lowing data  are  obtainable;  but  quantitatively  they  are  too  crude,  because 
they  ignore  the  essential  feature  of  oblique  reflection  from  M  and  N. 

Omitting  these  equations,  to  be  fully  discussed  in  the  next  chapter,  an 
example  of  the  displacement  and  radial  motion  in  a  given  experiment  may 
be  adduced;  the  former  was  fully  16  times  less  sensitive  per  fringe  than  the 
latter.  The  displacement  is  thus  a  coarse  adjustment  in  comparison  with 
the  usual  radial  motion  of  the  fringes  and  this  is  the  distinct  advantage  of 
the  present  method  for  many  purposes.  It  is  like  a  scale  division  into 
smaller  and  larger  parts,  where  the  enumeration  of  small  parts  alone  would 
be  confusing  or  impossible.  The  ratio  of  the  micrometer  value  of  displace- 
ment and  radial  motion  per  fringe  may  be  given  any  value  since  dz/d\  oc  e 
and  the  lateral  displacement  may  actually  be  more  sensitive  than  the 
radial  motion,  when  the  grating  plate  is  thin. 

The  sodium  lines  here  make  admirable  cross-hairs  and  the  ocular  itself 
need  have  none.  The  conditions  are  the  same  in  the  second  order  and  the 
coarser  rings  and  spectrum  lines  are  often  easier  to  count. 

34.  Compensator. — It  is  not  necessary,  however,  to  use  thin  glass,  for 
if  a  compensator  is  provided,  i.e.,  if  the  grating  is  on  the  common  plane 
between  two  thicknesses  of  identical  plane-parallel  glass  plates,  one  of 
which  carries  the  grating,  the  ideal  plane  in  question  is  provided.  The 
ellipses  in  this  case  would  be  infinite  in  size  and  their  displacements  infi- 
nitely large.  By  partial  compensation  (compensator  thinner)  ellipses  of 
any  convenient  size  and  rate  of  displacement  may  therefore  be  provided 
at  pleasure.  The  following  table  gives  some  rough  experimental  data 
where  z  is  the  advance  of  the  micrometer  screw  normal  to  the  grating,  i.e., 
the  displacement  of  the  grating  to  move  the  center  of  fringes  from  the  D  to 

TABLE  10. — Displacement  of  ellipses  from  the  D  to  the  b  line,  by  moving  the  grat- 
ing Az  cm.  normal  to  itself.  t  =  45°,  nearly.  Grating:  e  =  .6S  cm.,  D  => . 000351 
cm.,  ^=1.53  (estimated).  Compensation  shown  by  negative  sign.  Position  z 
taken  while  the  sodium  line  was  the  major  axis. 


. 

Com- 
pensator 
thickness* 
e'Xio2 

Position 
of  grat- 
ing z 

Microme-       NQ  of 

placement      ««*» 
A0Xio<         Diob 

Displace- 
ment per 
fringe 
io*Xdz/dn 

Ellipses. 

Side  of 
compen- 
sator. 

cm. 

cm. 

cm. 

cm. 

-48 

•°5 

30                         12 

2-5 

Very  large.  .  .  . 

N 

-44 

.08 

35 

Very  large  .... 

N 

-29 

.  12 

45 

Large  

N 

—  10 

.20 

65 

Mean  

N 

+  48 

.38 

US                   44 

2.6 

Very  small  .  .  . 

M 

-65 

.00 

10                    4.5 

2.3 

Eno'rmous.  .  .  . 

N 

±    o 

70                  28 

2.5 

Mean  

None 

*  Compensator  parallel  to  mirror,  M  or  N,  respectively. 


48  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

the  b  lines  of  the  spectrum,  e'  the  thickness  of  the  compensator  placed 
parallel  to  the  mirror  M  or  N  as  stated. 

Rotation  of  the  compensator  offers  the  usual  easy  method  of  adjustment. 
In  the  second  order  the  first  rings,  on  compensation,  usually  more  than 
fill  the  field.  This  is  of  course  eventually  the  case  in  the  first  order  also. 
With  very  perfect  compensation  the  ellipses  are  quite  eccentric  and  the 
lines  under  the  limiting  conditions  nearly  vertical  and  straight.  Hence  their 
motion,  partaking  of  the  twofold  character  specified,  is  complicated  but 
usually  opposite  in  direction  on  the  two  sides  of  the  center  for  the  same 
micrometer  displacement.  The  whole  phenomenon  may  vanish  within  a 
half  millimeter  of  play  of  the  grating,  passing  from  fine  lines  through  enor- 
mous ellipses  back  into  reversed  fine  lines,  all  nearly  vertical. 

The  displacement  of  the  grating  by  the  micrometer  screw  is  of  the  same 
order  per  fringe,  no  matter  whether  the  ellipses  are  large  or  small,  and  in 
the  last  table  it  was  about  .00025  cm.  per  fringe.  The  displacement  at  the 
mirror  would  exceed  this.  The  radial  motion  per  fringe  is  of  the  order  of 
wave-length.  Naturally  the  position  z  of  the  grating  changes  parallel  to 
itself  linearly  with  the  thickness  e'  of  the  compensator,  supposing  other  con- 
ditions the  same,  so  that  p,  z,  and  dz/dn  all  vary  linearly  with  e'  the  thick- 
ness of  the  compensator. 

The  full  equations  for  the  amounts  of  displacement,  etc.,  require  an 
evaluation  of  dQ/dn,  which  in  turn  must  take  into  consideration  that 
reflection  from  the  mirrors  can  not  in  general  be  normal  except  for  the  one 
color  instanced  above.  This  investigation  will  have  to  be  reserved  for  the 
next  chapter. 


CHAPTER   V. 


INTERFEROMETRY  WITH   THE  AID  OF  A  GRATING ;    THEORETICAL 
RESULTS.     By  Carl  Barus. 

PART  I.     INTRODUCTION. 

35.  Remarks  on  the  phenomena.— In  the  earlier  papers*  I  described 
certain  of  the  interferences  obtained  when  the  oblique  plate  gg,  fig.  18,  of 
Michelson's  adjustment,  is  replaced  by  a  plane  diffraction  grating  on 
ordinary  plate  glass.  Some  explanation  of  these  is  necessary  here.  In  the 
figure  L  is  the  source  of  white  light  from  a  collimator.  Such  light  is  there- 
fore parallel  relative  to  a  horizontal  plane,  but  convergent  relatively  to  a 
vertical  plane.  M  and  N  are  the  usual  silver  mirrors.  A  telescope  adjusted 
for  parallel  rays  in  the  line  GE  must  therefore  show  sharp  white  images  of 
the  slit.  As  the  grating  is  usually  slightly  wedge-shaped,  there  will  be 
(normally)  four  such  images,  two  returned  by  M  after  reflection  from  the 
front  (white)  and  rear  face  (yellowish)  of  the  plate  gg,  and  two  due  to  N. 
There  will  also  be  two  other,  not  quite  achromatic,  slit  images  from  N  or  M, 
respectively,  due  to  double  diffraction  before  and  after  reflection.  These 
will  be  treated  below.  In  the  direction  GD  there  will  thus  be  a  correspond- 
ing number  of  diffraction  spectra,  more  or  less  coincident  in  all  their  parts, 
and  therefore  adapted  to  interfere  in  pairs  throughout  their  extent.  If  the 
two  white  and  the  two  yellow  images  of  the  slit  be  put  in  coincidence  and 
the  mirrors  M  and  N  are  adjusted  for  the  respective  reduced  or  virtual  path 
difference  zero,  the  interferences  obtained  are  usually  eccentric;  i.e.,  the 
centers  of  the  interference  ellipses  are  not  in  the  field  of  view.  The  effective 
reflection  in  each  of  these  cases  takes  place  from  the  front  and  rear  face  of 
the  grating  at  the  same  time.  Hence  the  interference  pattern  includes  the 
prism  angle  of  the  grating  plate  and  is  not  centered.  The  air-paths  of  the 
component  rays  are  here  practically  equal.  In  addition  to  the  ellipses, 
this  position  also  shows  revolving  linear  interferences  and  (as  a  rule)  a 
double  set  is  in  the  field  at  once,  consisting  of  equidistant,  symmetrically 
oblique  crossed  lines,  passing  through  horizontality  in  opposite  directions 
together,  when  either  mirror  M  or  N  is  slightly  displaced. 

If  either  pair  of  the  white  and  yellowish  images  of  the  slit  be  placed  in 
coincidence  when  looking  along  EG,  the  interference  pattern  along  DG  is 
ring-shaped,  usually  quasi-elliptic  and  centered.  The  light  returned  by 
M  and  N  is  in  this  case  reflected  from  the  same  face  of  the  grating,  either 
from  the  face  carrying  the  grating  or  the  other  (unruled)  face.  The  corre- 
sponding air-paths  of  the  rays  are  in  this  case  quite  unequal,  because  the 

*  Am.  Journ.  of  Science,  xxx,  1910,  pp.  161-171;  Science,  July  15,  1910,  p.  92; 
and  Phil.  Mag.,  July,  1910,  pp.  45-59. 


50 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


short  air-path  is  compensated  by  the  path  of  the  rays  within  the  glass  plate. 
Hence  these  adjustments  are  very  different,  GM  being  the  long  path  in  one 
instance,  GN  in  the  other  instance.  For  the  same  motion  of  the  micrometer 
screw,  the  fringes  are  displaced  in  opposite  directions.  In  one  adjustment 
there  may  be  a  single  family  of  ellipses;  in  the  other  there  may  be  two  or 
even  three  families  nearly  in  the  field  at  once. 

If  the  grating  were  cut  on  optical  plate  glass,  the  adjustment  for  equal 
air-path  would  probably  be  best.  But  with  the  grating  cut  as  usual  on  ordi- 
nary plate,  or  in  case  of  replica  gratings  on  collodion  or  celluloid  films,  the 
adjustment  for  unequal  paths  is  preferable.  Here  again  one  of  the  positions 
is  much  to  be  preferred  to  the  other,  owing  to  the  occurrence  of  multiple 
slit  images  from  one  of  the  mirrors,  as  above  specified.  In  fig.  19,  for 
instance,  where  the  grating  face  is  to  the  rear,  there  are  but  two  images, 
i  and  2  from  M,  if  the  plate  is  slightly  wedge-shaped;  but  from  N,  in 


FIG.  1 8. — Diagram  showing  adjustments 
for  interference. 


FIG.  19. — Diagram  showing  double- 
diffraction. 


addition  to  these  two  normal  cases  (not  necessarily  coinciding  with  i  and  2), 
there  are  two  other  images  3'  and  4'  (3  and  4  are  spectrum  rays),  resulting 
from  double  diffraction,  with  a  deviation  6  and  angle  of  incidence  7, 
respectively  6  <  I  and  6  >  I,  in  succession ;  or  the  reverse.  As  the  compensa- 
tion for  color  can  not  here  be  perfect,  the  two  slit  images  obtained  are  very 
narrow,  practically  linear  spectra,  but  they  are  strong  enough  to  produce 
interferences  like  the  normal  images  of  the  slits,  with  which  they  nearly 
agree  in  position.  Other  very  faint  slit  images  also  occur,  but  they  may 
be  disregarded.  The  doubly-diffracted  slit  images  are  often  useful  in  the 
adjustments  for  interference. 

Among  the  normal  slit  images  there  are  two,  respectively  white  and  yel- 
lowish, which  are  remote  from  secondary  images.  If  these  be  placed  in  coin- 
cidence both  horizontally  and  vertically  along  EG,  fig.  18,  the  observation 
along  DG  through  the  telescope  will  show  a  magnificent  display  of  black 


IN    RELATION    TO    INTERFEROMETRY.  51 

apparently  confocal  ellipses,  with  their  axes  respectively  horizontal  and 
vertical,  extending  through  the  whole  width  of  the  spectrum,  from  red  to 
violet,  with  the  Fraunhofer  lines  simultaneously  in  focus.  The  vertical 
axes  are  not  primarily  dependent  on  diffraction  and  are  therefore  of  about 
the  same  angular  length  throughout ;  the  horizontal  axes,  however,  increase 
with  the  magnitude  of  the  diffraction,  and  hence  these  axes  increase  from 
violet  to  red,  from  the  first  to  the  second  and  higher  orders  of  spectra,  and 
in  general  as  the  grating  space  is  smaller.  It  is  not  unusual  to  obtain 
circles  in  some  parts  of  the  spectrum,  since  ellipses  which  in  one  extreme 
case  have  long  axes  vertically,  in  the  other  extreme  case  have  long  axes 
horizontally.  The  interference  figure  occurs  simultaneously  in  all  orders 
of  spectra,  and  it  is  interesting  to  note  that,  even  in  the  chromatic  slit  images 
shown  in  fig.  19,  needle-shaped  vertical  ellipses  are  quite  apparent. 

It  is  surprising  that  all  these  interferences  may  be  obtained  with  replica 
or  film  gratings,  though  not  of  course  so  sharply  as  with  ruled  gratings,the 
ideal  being  an  optical  plate.  With  thin  films  two  sets  of  interferences  are 
liable  to  be  in  the  field  at  once  and  I  have  yet  to  study  these  features  from 
the  practical  point  of  view.  If  the  film  is  mounted  between  two  identical 
plates  of  glass,  rigorously  linear,  vertical  and  movable  interference  fringes, 
as  described*  by  my  son  and  myself,  may  be  obtained. 

36.  Cause  of  ellipses.— The  slit  at  L,  fig.  18,  furnishes  a  divergent  pencil 
of  light  due  (at  least)  to  its  diffraction,  the  rays  becoming  parallel  in  a 
horizontal  section  after  passing  the  strong  lens  of  the  collimator.  But  the 
vertical  section  of  the  issuing  pencil  is  essentially  convergent.  Hence  if 
such  a  pencil  passes  the  grating  the  oblique  rays  relatively  to  the  vertical 
plane  pass  through  a  greater  thickness  of  glass  than  the  horizontal  rays. 
The  interference  pattern,  if  it  occurs,  is  thus  subject  to  a  cause  for  contrac- 
tion in  the  former  case  that  is  absent  in  the  latter.  Hence  also  the  vertical 
axes  of  the  ellipses  are  about  the  same  in  all  orders  of  spectra.  They  tend 
to  conform  in  their  vertical  symmetry  to  the  regular  type  of  circular  ring- 
shaped  figure  as  studied  by  Michelson  and  his  associates  and  more  recently 
by  Feussner.f  but  in  view  of  the  slit  the  symmetry  is  cylindric. 

On  the  other  hand  the  obliquity  in  the  horizontal  direction,  which  is 
essential  to  successive  interferences  of  rays,  is  furnished  by  the  diffraction 
of  the  grating  itself,  as  the  deviation  here  increases  from  violet  to  red.  In 
other  words  the  interference  which  is  latent  or  condensed  in  the  normal 
white  linear  image  of  the  slit  is  drawn  out  horizontally  and  displayed  in  the 
successive  orders  of  spectra  to  right  and  left  of  it.  The  vertical  and  hori- 
zontal symmetry  of  ellipses  thus  follows  totally  different  laws,  the  former 
of  which  have  been  thoroughly  studied  The  present  paper  will  therefore 
be  devoted  to  phenomena  in  the  horizontal  direction  only. 


*  Phil.  Mag.,  I.e. 

t  See  Professor  Feussner's  excellent  summary  in  Winkelmann's  Handbuch  der 
Physik,  vol.  6,  1906,  p.  958  et  seq. 


52  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

At  the  center  of  ellipses  the  reduced  path  difference  is  zero;  but  it  can  not 
increase  quite  at  the  same  rate  toward  red  and  violet.  Neither  does  the 
refractive  index  of  the  glass  admit  of  this  symmetry.  Hence  the  so-called 
ellipses  are  necessarily  complicated  ovals,  but  their  resemblance  to  confocal 
ellipses  is  nevertheless  so  close  that  the  term  is  admissible.  This  will  appear 
in  the  data. 

If  either  the  mirror  or  the  grating  is  displaced  parallel  to  itself  by  the 
micrometer  screw  the  interference  figure  drifts  as  a  whole  to  the  right  or  to 
the  left,  while  the  rings  partake  of  the  customary  motion  toward  or  from  a 
center.  The  horizontal  motion  in  such  a  case  is  of  the  nature  of  a  coarse 
adjustment  as  compared  with  the  radial  motion,  a  state  of  things  which  is 
often  advantageous — in  other  words,  the  large  divisions  of  the  scale  are  not 
lost ;  moreover  the  displacements  may  be  used  independently. 

The  two  motions  are  coordinated,  inasmuch  as  violet  travels  toward  the 
center  faster  in  a  horizontal  direction,  i.e.,  at  a  greater  angular  rate,  than 
red.  Hence  the  ellipses  drift  horizontally  but  not  vertically.  Naturally 
in  the  two  positions  specified  above  for  ellipses,  the  fringes  travel  in  opposite 
directions  for  the  same  motion  of  the  micrometer  screw.  As  the  thickness 
of  the  grating  is  less  the  ellipses  will  tend  to  open  into  vertical  curved  lines, 
while  their  displacement  is  correspondingly  increased.  With  the  grating 
on  a  plate  of  glass  about  e=.6&  cm.  thick  and  having  a  grating  space  of 
about  D  =  .000351  cm.,  at  an  angle  of  incidence  of  about  45°,  the  displace- 
ment of  the  center  of  ellipses  from  the  D  to  the  E  line  of  the  spectrum 
corresponded  to  a  displacement  of  the  grating  parallel  to  itself  of  about 
.006  cm.  It  makes  no  difference  whether  the  grating  side  or  the  plane  side 
of  the  plate  is  toward  the  light  or  which  side  of  the  grating  is  made  the  top. 
If  the  grating  in  question  is  stationary  and  the  mirror  N  alone  moves 
parallel  to  itself  along  the  micrometer  screw,  a  displacement  of  N  =  .01  cm. 
roughly  moves  the  center  of  ellipses  from  D  to  E,  as  before.  This  displace- 
ment varies  primarily  with  the  thickness  of  the  grating  and  its  refraction. 
It  does  not  depend  on  the  grating  constant.  Thus  the  following  data  were 
obtained  with  film  gratings  on  different  thicknesses  e  of  glass  and  different 
grating  spaces  D  for  the  displacements  N  of  the  mirror  at  N,  to  move  the 
ellipses  from  the  D  to  the  E  line,  as  specified. 

Glass  grating,  ruled e  =  .68  cm.    \/D  =  .168    N  =  .010  cm. 

Film  on  glass  plate e  =  .57  .352  .008 

Film  on  glass  plate e  =  .24  .352  .003 

Film  between  glass  plates.  •]    ,_'  .352  .003 

(  e  —  .24  / 

Reduced  linearly  to  e=.6S  cm.,  the  latter  data  would  be  AT  =  .oio  and 
W  =  .ooo,  which  are  of  the  same  order  and  as  close  as  the  diffuse  interfer- 
ence patterns  of  film  gratings  permit.  The  large  difference  in  dispersion, 
together  with  some  differences  in  the  glass,  has  produced  no  discernible 
effect. 


IN    RELATION    TO    INTERFEROMETRY. 


53 


An  interesting  case  is  the  film  grating  between  two  equally  thick  plates  of 
glass.  With  this,  in  addition  to  the  elliptical  interferences  above  described, 
a  pattern  of  vertical  interferences  identical  with  those  discussed  in  a  pre- 
ceding paper*  was  obtained.  These  are  linear,  persistently  vertical  fringes 
extending  throughout  the  spectrum  and  within  the  field  of  view,  nearly 
equidistant  and  of  all  colors.  Their  distances  apart,  however,  may  now  be 
passed  through  infinity  when  the  virtual  air-space  passes  through  zero ;  and 
for  micrometer  displacements  of  mirror  in  a  given  direction,  the  motion  of 
fringes  is  in  opposite  directions  on  different  sides  of  the  null  position  of  the 
mirror.  I  have  not  been  able,  however,  to  make  them  as  strong  and  sharp 
as  they  were  obtained  in  the  paper  specified. 


FIG.   20. — Diagram  showing  displace- 
ment of  mirror  N. 


FIG.  21. — Diagram  showing  displacement  of 
grating  G. 


37.  The  three  principal  adjustments  for  interference.  —  To  compute 
the  extreme  adjustments  of  the  grating  when  the  mirror  N  is  moved, 
fig.  20  may  be  consulted.  Let  ye  be  the  air-path  on  the  glass  side,  whereas 
y&  is  the  air-path  on  the  other,  e  the  thickness  of  the  grating  and  n  its  index 
of  refraction  for  a  given  color.  Then  for  the  simplest  case  of  interferences, 
in  the  first  position  N  of  the  mirror,  if  /  is  the  normal  angle  of  incidence  and 
ft  the  normal  angle  of  refraction  for  a  given  color, 


for  equal  paths.    Similarly  in  the  second  position  of  the  mirror 

y*=yt+en/cos  R 
Hence  if  the  displacement  at  the  mirror  N  be  N, 

N  =y*-yK  =  2  en/  cos  R+yK'-  y» 
The  figure  shows 

y*—  yK'  =  2e  tan  R  sin  / 
whence 


*  Phil.  Mag.,  I.e. 


54 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


The  measurement  of  this  quantity  showed  about  1.88  cm.  The  computed 
value  would  be  2  X  .68  X  i  .53  X  .7  1  =  i  .84  cm.  The  difference  is  due  to  the 
wedge-shaped  glass  which  requires  a  readjustment  of  the  grating  for  the 
two  positions. 

The  corresponding  extreme  adjustments,  when  the  grating  gg  instead  of 
a  mirror  N  is  moved  over  a  distance  z,  are  given  in  fig.  2  1  .  The  mirrors 
M  and  N  are  stationary.  It  is  first  necessary  to  compute  y,  the  angle 
between  the  displacement  z  and  the  side  5  in  the  figure,  or 

tan  y  =  tan  I  —  2e  tan  R/z 
so  that  the  semi-air-path  difference  of  rays  to  the  mirror  M  becomes 

s  =  y*—yK'  =  z(cos  7  —  sin  7  tan  7) 

We  may  now  write,  with  the  same  notation  as  before,  for  the  extreme  posi- 
tion of  the  grating,  G  and  G',  respectively, 

R 


2/cos  /  =  2?/i/cos  R  —  z(cos  /  —  sin  7  tan  7) 
whence 

2(1  /cos  7-f  cos  7—  sin  7  tan  7)  =  ze^/cos  R 

If  e  =  o,  it  follows  that  z  =  o,  or  7  =  7,  since  the  parenthesis  by  the  above 
equation  is  not  zero.  There  is  only  one  position.  The  angle  7  may  now  be 
eliminated  by  inserting  tan  y,  whence 


The 


_  ..      .-- 
cos  /  cos  R 

The  observed  value  of  z  for  the  two  positions  was  about  1.3  cm. 
computed  value  for  7  =  45°  is  the  same. 

On  the  other  hand,  when  reflection  takes  place  from  the  same  face  of  the 
grating,  while  the  latter  is  displaced  z  cm.  parallel  to  itself,  the  relations 
of  y  and  z  for  normal  incidence  at  an  angle  7  are  obviously 

y  cos  7  =  z 


FIG.   22. — Diagram  showing  displace-       FIG.   23. — Diagram  showing  displacement  of 
ment  of  grating  for  deviated  rays.  mirror  N  for  different  colors. 


IN    RELATION    TO    INTERFEROMETRY.  55 

For  an  oblique  incidence  t,  where  i  —  I=a,  a  small  angle,  the  equation 
is  more  complicated.  Fig.  22,  for  e  =  o,  g  and  g'  being  the  planes  of  the 
grating,  shows  that  in  this  case 

sin  a   .     ,.  .        7 
y  =  z(i+2  ----  -.  sin  /)/cos  / 
cos  * 

or,  approximately  for  small  a 

y  =  z(i+2  sin  a  tan  7)/cos  7 

This  equation  is  also  true  for  a  grating  of  thickness  e,  whose  faces  are  plane 
parallel,  for  the  direction  of  the  air-rays  in  this  case  remains  unchanged. 
Finally,  a  distinction  is  necessary  between  the  path  difference  2y  and  the 
motion  of  the  opaque  mirror  2N,  which  is  equivalent  to  it,  since  the  light 
is  not  monochromatic.  This  motion  2  A7  is  oblique  to  the  grating  and  if 
the  rays  differ  in  color  further  consideration  is  needed.  In  fig.  23  for  the 
simplest  type  of  interferences,  in  which  the  glass-path  difference  is  en/cos  R 
and  for  normal  incidence  at  an  angle  I,  let  the  upper  ray  yu  and  the  grating 
be  fixed.  The  mirror  moving  over  the  distance  JN  changes  the  zero  of  path 
difference  from  any  color  of  index  of  refraction  HD  to  another  of  index  ME, 
while  ;yND  passes  to  ;yNE.  Then 

y*l  =  ;VND  +  £,«D  COS  #D  = 

Hence 

J.V  =  -esin  7  (tan  #t)-tan 

COS 


or 

JN  =e(fJLD  COS  RD  —  fJLE  COS  .RE) 

This  difference  belongs  to  all  rays  of  the  same  color  difference,  or  for  two 
interpenetrating  pencils. 

If  reflection  takes  place  from  the  lower  face  the  rays  are  somewhat  dif- 
ferent, but  the  result  is  the  same.  If  the  plate  of  the  grating  is  a  wedge  of 
small  angle^,  the  normal  rays  will  leave  it  on  one  side  at  an  angle  7+5 
where  5  is  the  deviation 


The  mirror  N  will  also  be  inclined  at  an  angle  7+5,  to  return  these  rays 
normally.    We  may  disregard  d8  =  <pdn,  if  «p  is  small. 

PART  II.  DIRECT  CASES  OF  INTERFERENCE.  DIFFRACTION  ANTECEDENT. 
38.  Diffraction  before  reflection.—  If  in  fig.  18  the  diffracted  beams 
(spectra)  be  returned  by  the  mirrors  M'  and  N'  to  be  reflected  at  the  grating, 
interference  must  also  be  producible  along  GD.  Again  there  will  be  three 
primary  cases:  if  reflection  takes  place  from  both  faces  of  the  grating  at 
once,  the  air-paths  must  be  nearly  equal,  the  grating  itself  acting  as  a 
compensator.  The  interference  pattern  is  ring-shaped,  but  as  usual  very 
eccentric.  If  the  reflection  of  both  component  beams  takes  place  from  the 


56 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


same  face  of  the  grating,  the  interference  pattern  is  elliptical  and  centered 
and  the  air-paths  are  unequal.  The  case  is  then  similar  to  the  preceding 
§§i,  2;  but  there  is  no  direct  or  normal  image  of  the  slit,  as  M  and  N  are 
absent.  In  its  place  there  may  be  chromatic  images  of  the  slit  (linear 
spectra)  at  C  due  to  the  double  diffraction  of  each  component  beam  in  a 
positive  and  negative  direction  successively.  But  these  chromatic  images 
are  nevertheless  sharp  enough  to  complete  the  adjustments  for  interference 
by  placing  the  slit  images  in  coincidence.  It  is  not  usually  necessary  to  put 
the  spectrum  lines  into  coincidence  separately  (both  horizontally  and  ver- 
tically), as  was  originally  done,  both  spectra  being  observed.  Again  along 
GE,  fig.  1 8,  approximately,  there  must  be  two  successive  positive  diffrac- 
tions of  each  component  beam,  which  would  correspond  closely  to  the 
second  order  of  diffraction.  The  advantage  of  this  adjustment  lies  in  the 
fact  that  there  are  but  two  slit  images  effectively  returned  by  M'  and  N't 
and  hence  these  interferences  were  at  first  believed  to  be  stronger  and  more 
isolated.  As  a  consequence  I  used  this  method  in  most  of  my  early  experi- 
ments, before  finding  the  equally  good  adjustment  described  in  the  preced- 
ing section. 


FIG.   24. — Diagrams  showing  interferences  of  diffracted  rays,  case  H  normal  to 
mirrors  M  and  N;  case  0  oblique  to  mirrors.     Double  incidence. 

39.  Elementary  theory.— To  find  the  path  differences  figs.  24  and  25 
may  be  consulted.  In  both  the  grating  face  is  shown  at  gg,  the  glass 
plate  being  c  cm.  in  thickness  below  it,  and  n  is  the  normal  to  the  grating. 
M  and  N  are  the  two  opaque  mirrors,  each  at  an  angle  9  to  the  face  of  the 
grating.  Light  is  incident  on  the  right  at  an  angle  *  nearly  45°.  In  both 
figures  the  rays  ym  and  >•„  (air-paths)  diffracted  at  an  angle  S  in  air,  are 
reflected  normally  from  the  mirrors  M  and  N  respectively  and  issue  toward 
p  for  interference.  The  rays  yn  (primed  in  figure)  pass  through  glass. 
Both  figures  also  contain  two  component  rays  diffracted  at  an  angle  6  in 
air,  where  6—  8  =  a,  and  reflected  obliquely  at  the  mirrors  M  and  N,  thus 
inclosing  an  angle  20.  in  air  and  2/3,  in  glass,  and  issuing  toward  q.  These 


IN    RELATION    TO    INTERFEROMETRY. 


57 


component  rays  are  drawn  in  full  and  dotted,  respectively.  In  fig.  24  there 
are  two  incident  rays  for  a  single  emergent  ray,  in  fig.  25  a  single  incident 
ray  for  two  emergent  rays  interfering  in  the  telescope.  The  treatment  of 
the  two  cases  is  different  in  detail;  but  as  the  results  must  be  the  same, 
they  corroborate  each  other. 


A 


FIG.  25. — Diagrams  showing  interferences  of  diffracted   rays,  case  9  normal  to 
mirrors  M  and  N;  case  6  oblique  to  mirrors.     Single  incidence. 

The  notation  used  is  as  follows:  Let  e  be  the  normal  thickness  of  the 
grating,  e'  the  effective  thickness  of  the  compensator  when  used.  Let  dis- 
tance measured  normal  to  the  mirror  be  termed  y,yn  and  ym  being  the  com- 
ponent air-paths  passing  on  the  glass  and  on  the  air  side  of  the  grating  gg, 
so  that  ym>yn.  Let  y=ym— y*  be  the  air-path  difference.  Similarly  let 
distances  z  be  measured  normal  to  the  grating,  so  that  z  may  refer  to  dis- 
placements of  the  grating. 

Let  i  be  the  angle  of  incidence,  r  the  angle  of  refraction,  and  /ur  the  index 
of  refraction  for  the  given  color,  whence  sin  i  =  HT  sin  r.  Similarly  if  8  is  the 
angle  of  diffraction  in  air  of  the  y  rays  and  S\  the  corresponding  angle  in 
glass, 

sin  #  =  («esin  Sl  (i) 

and  if  6  is  the  angle  of  diffraction  of  any  oblique  ray  in  air  and  8\  the  cor- 
responding angle  in  glass 

sin  0  =  sin  (0+a)  =  /*«  sin  0,  (2) 

I  have  here  supposed  that  d>6,  so  that  a  =  0—9  is  the  deviation  of  the 
oblique  ray  from  the  normal  ray  in  air.  Finally,  the  second  reflection  of 
the  oblique  ray  necessarily  introduces  the  angle  of  refraction  within  the 
glass  such  that 

sin  (0- a)- /*«  sin  ft  (3) 

If  D  is  the  grating  space  we  have,  moreover, 

sin  i  —  sin  0  =  X0jD    and     sin  i  —  sin  0  =  Je/Z>  (4) 


58  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

For  a  displacement,  z,  of  the  grating,  or  an  equivalent  displacement,  y, 
of  either  mirror,  the  path  difference  produced  in  case  of  the  oblique  ray 
will  be 

2^/cos  a  =  22  (cos  (i  —  0)+sin  (i  —  9)) /cos  a  cos  i  (5) 

For  normal  rays  cos  a  =  i .  If  as  in  paragraph  i  the  reflection  is  ante- 
cedent 

i  =  0  2}/cos  a  =  22/cos  a  cos  i 

40.  Equations  for  the  present  case.— From  a  solution  of  the  triangles, 
preferably  in  the  case  for  single  incidence,  as  in  fig.  25,  the  air- path  of  the 
upper  oblique  component  ray  at  a  deviation  a  is 

2ym  cos  0/cos  (9—  a) 
The  air-path  of  the  lower  component  ray  is 

2  cos  0  (yn  —  e  sin#  (tan  6\  —  tan  6>i))/cos  (Q  —  a) 

The  optic  path  of  this  (lower)  ray  in  glass  as  far  as  the  final  wave  front  in 
glass  at  /3i  is 

jUf*(l/COSdt+ I/COS /3J 

The  optic  path  of  the  upper  ray  as  far  as  the  same  wave  front  in  glass  is 

/is  sin  ft  1 2 -£  -    .  (y,,,  —  yn  —  e  sin  9  (tan  0,  —  tan  0,) )  — 

e(tan0t-tanft)[        (6) 

Hence  the  path  difference  between  the  lower  and  the  upper  ray  as  far  as  the 
final  wave  front  is,  on  collecting  similar  terms, 

cos  0          sin  a  sin  (0-a) 
lS^  +    ^cos  (*--«) 


+  e  sin  (0  -  a)  (tan  0t  -  tan  0t)  -         ,2—     .  (tan  Ol  -  tan  0,  ) 

cos  (ff  —  a) 

(sin  0  cos  6  +sin  a  sin  9  sin  (0  —  a))  (7) 

Before  reducing  further,  one  may  remark  that  if  incident  and  emergent 
rays  were  coincident,  the  quantity  given  under  (6)  would  be  zero.  Hence 
if  this  particular  yn—ym  =  y\,  it  follows  that 

2y,  =  2e  sin  #(tan  0,  -  tan  0,)  -      °~  ^  (tan  0t  -  tan  ft) 


Hence  the  particular  path  difference  for  this  case  would  be 

/cos  0,  +  i  /cos  ft)  -e  cos  0(tan  0,-tan  ft)  /cos  a 


which  becomes  indeterminate  when  a  =  o. 

To  return  to  path  differences,  equation  (7),  the  coefficients  of  ym—  y*  =  y 
(where  y  is  positive)  and  of  e  may  now  be  brought  together,  as  far  as 


IN  RELATION  TO  INTERFEROMETRY. 


59 


possible.  If  the  path  difference  corresponds  to  n  wave-lengths  X«,  we  may 
write  after  final  reduction 

nXe=  —  2y  cos  a+  20/xe  cos  a(i/cos  61  —  cos  (Ol—  0X)  /cos  0t) 

+  efie(  i  +  cos  (0i  -  ft)  )/cos  ft         (8) 

which  is  the  full  equation  in  question  for  a  dark  fringe.  It  is  unfortunately 
very  cumbersome  and  for  this  reason  fails  to  answer  many  questions  per- 
spicuously. It  will  be  used  below  in  another  form. 

The  equation  refers  primarily  to  the  horizontal  axes  of  the  ellipses  only, 
as  e  increases  vertically  above  and  below  this  line.  The  equation  may  be 
abbreviated 

nXo  =  —  zy  cos  a  +  eZ®  +  eZ6 

where  Z0  and  Z&  are  the  coefficients  of  e  in  the  equation.  In  case  of  a  par- 
allel compensator  of  thickness  e'  we  may  therefore  write,  y  being  the  path 
difference  in  air, 

nX6  =  —  2y  cos'a  +  (e  —  e')  (Ze  —  Ze)  (9) 

If  e  =  e',  i.e.,  for  an  infinitely  thin  plate  e  =  o, 

(10) 


(n) 


(12) 


Again  for  normal  rays  ym  and  y  n,  a  =  o  and  0i  =  0i  =  ft.    Hence 


and  for  a  parallel  compensator  of  thickness  e' 


If  /xe=  i  =ne,  equation  (8)  reduces  to 

nX  =  2  cos  a(;y  -f  e/cos  #)  , 
clearly  identical  with  the  case  e  =  o  for  a  different  y. 


TABLE  n.  —  -Table  for  path  difference  ,  —  a(ym  —  ya)cosa+eZQ  +  eZg  =  - 
Light  crown  glass.  E  rays  normal  to  mirrors.  Grating  space,  D,  =• 

•aycoa  a+eZ. 
.000  3512  cm. 

7  =  45°.     0-33°  Si'- 

Spectrum  lines         = 

B 

D 

E 

F 

G 

AX  10'  = 

68.7 

58.93 

52.7° 

48.61 

43  .08  cm. 

9  = 

i  .5118 
30°  46' 

32°  38' 

i  .5186 
33°  Si' 

1  .5214 
34°  40' 

1.5267 

35°  46' 

0,  = 

19°  47' 

20    51' 

21    31' 

21     57' 

22      30' 

a  = 

-    3°     5' 

-    i    13' 

o      o' 

o    49' 

1    55' 

o\~&i  = 

-     i°44' 

-    o    40' 

o      o' 

o    26' 

o    59' 

PI  — 

23°  25' 

22      17' 

21      30' 

21      I3' 

20    49' 

0i—  &  = 

-    3°  38' 

-     I      26' 

o      o' 

o    44' 

i    41' 

2  cos  a  = 

1.9970 

1.9996 

2  .OOOO 

1.9998 

i  .  9996  cm. 

Ze  = 

-0383 

.0176 

.OOOO 

-   .0095    - 

.0219  cm. 

Z&  = 

3.2100 

3  •  2404 

3  •  2648 

3  -2805 

3  .3038  cm. 

Path  difference  *     n/  =  \ 

-1.9970) 

—  i  .99963 

>-2.  0000) 

-i.ggg&y- 

1.9996)-  )  cm 

( 

+  3  •  24830 

+  3  .2580* 

+  3  •  26486 

+  3.2710*?  + 

3  .  2819?  J 

Path   difference) 
"=3  .  2648  cm.  >•  2Ay0=    —    .0116    — 


.0062    ±     .0000+    .0059  +      .0177  cm. 


The  same  for  ) 
<?=.68cm.  / 


=    —      0079   —    .0042    ± 
=    —     .0107    -    .0056    ± 


.0040  + 
.0049+ 


.oi2ocm. 
.oi56cm 


*  The  purpose  of  these  data  is  merely  to  elucidate  the  equations.     AAT  refers  to 
the  displacement  of  either  opaque  mirror,  M  or  N. 


60 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


In  view  of  the  complicated  nature  of  equation  (8) ,  I  have  computed  path 
differences  for  a  typical  case  of  light  crown  glass,*  as  shown  in  table  1 1,  for 
the  Fraunhofer  lines  B,  D,  E,  F,  G,  supposing  the  E  ray  to  be  normal  (a  =  o). 
The  table  is  particularly  drawn  up  to  indicate  the  relative  value  of  the 
functions  Ze  and  Ze.  It  will  be  seen  that  Z  =  Ze-fZfl  in  terms  of  X,  is  a 
curve  of  regular  decrease  having  no  tendency  to  assume  maximum  or 
minimum  values  within  the  range  of  X,  whereas  y  cos  a  passes  through  a  flat 
maximum  for  a  =  o.  This  is  shown  in  the  curves  of  fig.  26. 


3*60 


50 


52 


48 


£ 


-  -X  X 10 


46      48       50     5£       545658606264 


76— if 

FIG.  26. — Chart  showing  path  difference  in  centimeters  (to  be  annulled 
at  E  line),  in  terms  of  wave-length. 

The  path  difference,  which  is  the  difference  of  the  ordinates  of  these  curves, 
thus  passes  through  zero  for  a  definite  value  of  e  and  y,  and  this  would 
at  first  sight  seem  to  correspond  to  the  center  of  ellipses.  That  it  does  not 
so  correspond  will  be  particularly  brought  out  in  the  next  section,  but  here 
this  tentative  surmise  may  be  temporarily  admitted  to  simplify  the  descrip- 
tions. It  is  obvious  that  for  a  given  e  the  value  of  y  (air-path  difference) 
which  makes  the  total  path  difference  zero,  varies  with  the  wave-length, 
hence  on  increasing  y  continuously  the  ellipses  must  pass  through  the 
spectrum.  It  is  also  obvious  that,  if  the  grating  is  reversed,  the  path 
difference  will  change  sign,  caet.  par.,  and  the  ellipses  will  move  in  a  con- 
trary direction,  for  the  same  displacements  y  of  the  micrometer  screw, 
at  the  mirror  M  or  N,  respectively. 

If  e=  i  cm.  and  ^  =  3.2648  cm.,  the  path  difference  will  be  zero  for  the  E 
ray  and  (barring  further  investigation  as  just  stated)  one  is  inclined  to  place 
the  center  of  ellipses  there.  The  table  shows  the  residual  path  differences 
in  the  red  and  blue  parts  of  the  spectrum.  Fig.  27  has  been  constructed  for 


*  Taken  from  Kohlrausch's  Practical  Physics,  nth  edition,  1910,  p.  712. 


IN    RELATION    TO    INTERFEROMETRY.  61 

the  same  condition.  It  follows,  therefore,  that  in  proportion  as  e  is  smaller 
y  will  be  smaller  for  extinction  at  the  E  line.  Hence  their  differences  will 
be  smaller  and  the  number  of  wave-lengths  corresponding  to  the  path 
difference  of  the  Fraimhofer  lines  will  be  smaller.  In  other  words,  the  major 
axes  of  the  ellipses  will  be  larger.  Thus  the  ellipses  will  be  larger  as  e  is 
smaller,  the  limit  of  enormous  ellipses  being  reached  for  e  =  o  or  e  =  e'  of  the 
compensator. 

This  result  may  be  obtained  to  better  advantage  by  reconstructing  equa- 
tion (8),  and  making  the  path  difference  equal  to  zero  for  the  normal  ray. 
This  determines 

sB  (13) 


in  terms  of  e,  and  the  path  difference  now  becomes 

nX9  —  e{  —  /«e  cos  a  cos  (0^—  &i)  +/ie(i  +  cos  (6l  —  &))}  /cos  0t       (14) 


This  is  zero  for  a  =  o,  Q\ =/3i  =  Si  at  the  E  line.  The  other  values  are  given 
in  the  table  in  centimeters  and  in  wave-lengths.  Path  difference  diminishes 
as  e;  the  horizontal  axes  of  the  ellipses  increase  with  e.  If  observations 
are  made  very  near  the  E  line  (normal  ray)  the  following  approximate  form 
may  be  used : 

but  this  rapidly  becomes  insufficient  unless  0i  —  S\  and  0i — /8i  are  very  small. 

4 1 .  Interferometer. — If  in  equation  (8),  y  alone  is  variable  with  the  order 
of  fringe  n  while  a,  6,  S,  Q\,  Q\,  /Si,  e,  X,  /*,  are  all  constant  (i.e.,  if  the  number 
of  fringes  n  cross  a  given  fixed  spectrum  line  like  the  D  line,  when  the  mirror 
is  displaced  over  a  distance  y)  it  appears  that 

dn  ~  2  cos  a 

where  a  refers  to  the  deviation  of  Xfl  from  Xe  normal  to  the  mirror.  If 
a  =  o,  dy/dn  —  o,  the  limiting  sensitiveness  of  the  apparatus,  which  appears 
for  the  case  of  normal  rays. 

If  the  grating  is  displaced  along  z 

dz  _     Xe  cos  i 

dn     2  cos  a  cos  (i  —  8)+  sin  (i  —  9} 

where  cos  a  =  i  for  normal  rays.  In  both  cases  the  displacement  per  fringe 
dy/dn  and  dz/dn  vary  with  the  wave-length.  Hence  if  the  ellipses  are 
nearly  symmetrical  on  both  sides  of  the  center,  i.e.,  if  the  red  and  violet 
sides  of  the  periphery  are  nearly  the  same,  or  the  fringes  nearly  equidistant, 
the  smaller  wave-length  will  move  faster  than  the  larger  for  a  given  dis- 
placement of  mirror.  In  other  words,  the  violet  side  will  pass  over  a  whole 
fringe  before  the  red  and  consequently  the  ellipses,  as  a  whole,  must  drift  in 


62 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


the  direction  in  which  violet  moves.  It  is  at  first  quite  puzzling  to  observe 
the  motion  of  ellipses  as  a  whole  in  a  direction  opposed  to  the  motion  of 
the  more  reddish  fringes  when  these  are  alone  in  the  field. 

42.  Discrepancy  of  the  table.— The  data  of  table  i  are  computed  sup- 
posing that  the  path  difference  zero  corresponds  to  the  center  of  ellipses. 
This  assumption  has  been  admitted  for  discussion  only  and  the  inferences 
drawn  are  qualitatively  correct.  Quantitatively,  however,  the  displace- 
ment of  about  JAf=.oc>3  cm.  should  move  the  center  of  ellipses  from  the 
D  to  the  E  line  of  the  spectrum,  whereas  observations  show  that  a  displace- 
ment of  either  opaque  mirror  of  about  JAr=.oi  cm.  is  necessary  for  this 
purpose;  i.e.,  over  three  times  as  much  displacement  as  has  been  computed. 
The  equation  can  not  be  incorrect ;  hence  the  assumption  that  the  center  of 
ellipses  corresponds  to  the  path  difference  zero  is  not  vouched  for  and  must 
be  particularly  examined.  This  may  be  done  to  greater  advantage  in  con- 
nection with  the  next  section,  where  the  conditions  are  throughout  simpler, 
but  the  data  are  of  the  same  order  of  value. 


FIG.  27. — Diagram  showing  interferences  of  reflected  rays, 
subsequently  diffracted.  Case  /,  normal  to  mirrors; 
case  i  =  7  +  a  oblique  to  mirrors.  Rays  R  refracted,  D 
diffracted. 

PART  III.     DIRECT  CASE;  REFLECTION  ANTECEDENT. 

43.  Equations  for  this  case.— If  reflection  at  the  opaque  mirrors  takes 
place  before  diffraction  at  the  grating,  the  form  of  the  equations  and  their 
mode  of  derivation  are  similar  to  the  case  of  paragraph  40,  but  the  variables 
contained  are  essentially  different.  In  this  case  the  deviation  from  the 


IN  RELATION  TO  INTERFEROMETRY.  63 

normal  ray  is  due  not  to  diffraction  but  to  the  angle  of  incidence,  and  the 
equations  are  derived  for  homogeneous  light  of  wave-length  X  and  index  of 
refraction  /*• 

In  fig.  27  let  y^—y'  and  ym  =  y  be  the  air-paths  of  the  component  rays, 
the  former  first  passing  through  the  glass  plate  of  thickness  e.  Let  the 
angle  of  incidence  of  the  ray  be  /,  so  that  ym  and  yn  are  returned  normally 
from  the  mirrors  M  and  N,  respectively,  these  being  also  at  an  angle  /  to 
the  plane  of  the  grating.  Let  i  be  the  angle  of  incidence  of  an  oblique  ray, 
whose  deviation  from  the  normal  is  *  —  7  =  a.  Let  R,  r,  and  ft  be  angles  of 
refraction  such  that 

sin«  =  sin  (J+a)  =/x  sin  r       sin/  =  /zsin.R        sin  (/—  a)=/*sin  ft    (18) 


The  face  of  the  grating  is  here  supposed  to  be  away  from  the  incident  ray, 
as  shown  at  gg  in  the  figure. 


TABLE   12.  —  Path  difference,  —  2(ym  —  yn)cosa  +  eZ1—eZi  +  eZ3.     Light  crown  glass. 
a  =  3°  throughout.     Grating  space  .000351  cm.     7  =  45°.     e  =  icm. 


Spectrum 

lines 

B 

D 

E 

F 

G 

/  ) 

<  10'  = 

68.7 

58-93 

52.70 

48.61 

43  .  08  cm. 

i: 

i  .5118 
27°  53' 

27°  49' 

1.5186 
27°  45' 

i  .5214 
27°  42' 

1.5267 
27°   35' 

r  = 

29    26 

29°  22' 

29°  18' 

29°  14' 

29°    8' 

A  = 

26°  16' 

26°  12' 

26°     8' 

26°    5' 

26°     o' 

a  = 

3°    o' 

T.°       O' 

3°    o' 

}°    o' 

3°    o' 

r 

-/?  = 

i°33' 

i°33' 

i°33' 

i°33' 

i°33' 

r 

~~A  = 

3°  10' 

3°  10' 

3°  10' 

3°     9' 

3°     8' 

z  = 

3  .4160 

3.4218 

3-4272 

3-43i8 

3  .4403  cm. 

Z2  = 

656 

767 

808 

896  cm. 

Z2  = 

689 

647 

799 

841 

928  cm. 

Path    difference, 
a  =  3°,  e  =  i  cm., 
y  =  i  cm. 

H 

-1.9992 
+  3-4193 

-1.9992 

+  3-4252 

-1.9992 
+  3-4304 

-1.9992 
+  3-4352 

-1.9992 
+  3-4436 

If  path  differ-  "I 

ence  is  annulled  ! 
at  £  line,  0  =  3°,  [ 

2Ay0- 

—      .0111 

-     .0052 

±  .0 

+    .0048 

+    .0132 

e  =  i  cm. 

J=C0o  S?™i'cmi}2*y°=     '"    -0111    ~    -0052       ±'°          +    -°°48    +    -0132 

The  data  are  merely  intended  to  elucidate  the  equations. 

The  values  Z  are  nearly  equal.    So  also  the  cases  for  0  =  3°  and  a  =  o°. 

Fig.  28  has  been  added  to  accentuate  the  symmetrical  conditions  when 
a  compensator  parallel  to  the  grating  and  of  the  same  thickness  is  employed. 

Then  it  follows  as  in  paragraph  40,  mutatis  mutandis,  that  the  path  dif- 
ference is  (if  y  =  ym  -  y  „) 

f  2  cos  a     2  cos  a  cos  (r  —  R)      i  +  cos  (r  —  8.)  \ 
n\=  -zycosa  +  fie  <  -      D-  -+- 

t  cos  R  cosr  cos  r          J  (19) 

=  —  2y  cos  a  +  0(Z,  —  Z2  +  Z3) 


64 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


where  n  is  the  order  of  interference  (whole  number).  This  equation  is 
intrinsically  simpler  than  equation  (8)  since  /ir  =  /u«,  as  already  stated,  and 
since  a  is  constant  for  all  colors,  or  all  values  of  /x  and  A,  in  question.  R  and 
r  replace  S\  and  B\. 

In  most  respects  the  discussion  of  equation  (19)  is  similar  to  equation  (8) 
and  may  be  omitted  in  favor  of  the  special  interpretations  presently  to  be 
given.  If  M=  i,  equation  (19)  like  equation  (8)  reduces  to  the  case  corre- 
sponding to  e  =  o,  with  a  different  y  normal  to  the  grating. 

All  the  colors  are  superposed  in  the  direct  image  of  the  slit,  Rn  and  R  in 
fig.  27,  seen  in  the  telescope,  and  the  slit  is  therefore  white.  This  shows 


FIG.   28. — Diagram  showing  symmetrical  interference  for  compen- 
sated grating,  gg.     Rays  oblique  to  mirror. 


also  that  prismatic  deviation  due  to  the  plate  of  the  grating  (wedge)  is  inap- 
preciable. The  colors  appear,  however,  when  the  light  of  the  slit  is  analyzed 
by  the  grating  in  the  successive  diffraction  spectra,  Dn  or  D,  respectively. 
In  equation  (19)  n  is  a  function  of  X  and  hence  of  the  deviation  6  produced 
by  the  grating,  since  sin  (7  — a)— sin  6=\/D. 

The  values  of  equation  (19)  for  successive  Fraunhofer  lines  and  for 
0  =  3°,  have  been  computed  in  table  1 2  for  the  same  glass  treated  in  table  1 1 , 
the  data  being  similar  and  in  fact  of  about  the  same  order  of  value.  The 
feature  of  this  table  is  the  occurrence  of  nearly  constant  values  o.=i—I, 


IN    RELATION    TO    INTERFEROMETRY. 


65 


r— R  and  r— ft,  throughout  the  visible  spectrum.  Hence  if  the  following 
abbreviations  be  used 

A  =  cos  a  =  .9986       B  =  cos  (r  —  R)  =  .9996       C  =  i +cos  (r  — /?,)  =  i  .9984 

A,  #,  C  are  practically  functions  of  a  only  and  do  not  vary  with  color  or  X. 
Furthermore,  if  the  path  difference  is  annulled,  at  the  E  line  for  instance, 
for  the  normal  ray,  since  r  =  ft  =  R,  a  =  o,  so  that  R  =  RE  is  the  normal  angle 
of  refraction, 

o  =  —  y  4-  2ne/cos  RE 

and  equation  (19)  reduces  to 

n\  =  ieA  (///cos  R  —  fj.  cos  RE)  +  e(C  —  2AB)lu/cos  r  (20) 

where  zeA  and  e(C—  2AB)  are  nearly  independent  of  X  and  the  last  quan- 
tity is  relatively  small.  The  two  terms  of  this  equation  for  e=  i  cm.  show 
about  the  following  variation: 

TABLE  13. 


B 

D 

E        F 

G 

—  .0111 

—  .000009 

—  .0052 
—  .  000004 

.0     +.0048 
.0     +  .000005 

.0132 

.000013 

Hence  for  deviations  even  larger  than  a  =  3°,  the  path  difference  does  not 
differ  practically  from  the  path  difference  for  the  normal  ray.  Thus  it 
follows  that  the  equation 

nX  =  —  2y  -f  2£///cos  R  (21) 

is  a  sufficient  approximation  for  such  purposes  as  are  here  in  view. 

Finally,  for  e  =  .68  cm.  (the  actual  thickness  of  the  plate  of  the  grating) , 
y  and  N,  the  semi-path  difference  and  the  displacement  of  the  opaque 

mirror,  will  be 

TABLE  14. 


B 

D 

E         F         G 

y0  = 
Ay0  = 

•Wo  = 
AJVo  = 

1.1630 
-  .0038 
+  .0014 
-  .0052 

i  .  1650 
—  .0018 
+  .0007 
-  .0025 

I.  1668             I.  1686           LI730 

.0        +  .0016     +  .0045 
.0         -  .0005     -  .0018 
.0        +  .0021     +  .0063 

where  5A/o  is  the  color  correction  of  dy0  and  JN0  =  dy0  —  dN0  determines 
the  corresponding  displacements  of  the  opaque  mirror;  or  more  briefly 


N0=y0—e/j.  tan  R  sin  R  =  zen  cos  R 
The  data  actually  found  were 

C  DBF 

JN0=    —.0148         —.0093         .o         +.0057 


(22) 


66  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

These  results  are  again  from  2  to  4  times  larger  than  the  computed  values. 
True  the  glass  on  which  the  grating  was  cut  is  not  identical  with  the  light 
crown  glass  of  the  tables;  but  nevertheless  a  discrepancy  so  large  and  irreg- 
ular is  out  of  the  question.  It  is  necessary  to  conclude  therefore  that  here, 
as  in  paragraph  42,  the  assumption  of  a  total  path  difference  zero  for  the 
center  of  ellipses  is  not  true.  In  other  words  the  equality  of  air-path  differ- 
ence and  glass-path  does  not  correspond  to  the  centers  in  question.  It  is 
now  in  place  to  examine  this  result  in  detail. 

44.   Divergence  per  fringe.—  The  approximate  sufficiency  of  the  equation 
n\=2€fM/cos  R—  2y  =  z€fjL  cos  R  —  2N  (23) 

makes  it  easy  to  obtain  certain  important  derivatives  among  which  dd/dn, 
where  6  is  the  angle  of  diffraction  corresponding  to  the  angle  of  incidence 
/  and  M  the  order  of  interference,  is  prominent. 

If  e  and  y  are  constant  and  if  n,  \,  M  and  R  are  variable,  the  differential 
coefficients  may  be  reduced  successively  by  the  following  fundamental 
equation,  D  being  the  grating  space,  M  and  r  corresponding  to  wave-length 
X  and  angle  of  diffraction  6. 

dR=-tanRd»/n  (24) 

d\=-D  cos  B  •  dd  (25) 

(26) 


where,  as  a  first  approximation,  a  =  .015,  is  an  experimental  correction, 
interpolated  for  the  given  glass.  Incorporating  these  equations  it  is  found 
that 

d0=__£  __  cosK 

dn     2Dcos0  en-ycosR+aefi(i-ta.n2  R)  *7' 


If  the  path  difference  n\  is  annulled 

ddQ  =      _£__  cos/? 

dn     2  D  cos  0  ae//(i-tan2/?) 


which  is  the  deviation  per  fringe,  supposedly  referred  to  the  center  of 
ellipses. 

These  equations  indicate  the  nature  of  the  dependence  of  the  horizontal 
axes  of  ellipses  on  i/D  (hence  also  on  the  order  of  the  spectrum),  i/e,  i/n, 
and  i/a,  where  the  meaning  of  a,  here  apparently  an  important  variable, 
is  given  in  equation  (26).  If  instead  of  the  path  difference  y  the  displace- 
ment N  of  either  opaque  mirror  is  primarily  considered  (necessarily  the 
case  in  practice),  the  factor  (i  —  tan2  R)  vanishes. 

Table  15  contains  a  survey  of  data  for  equations  (27)  and  (28).  The 
results  for  dB0/dn  would  be  plausible,  as  to  order  of  values.  The  data  for 
d$/dn,  however,  are  again  necessarily  in  error,  as  already  instanced  above, 
paragraph  42.  They  do  not  show  the  maximum  at  e  and  the  X—  effect  is 
overwhelmingly  large. 


IN    RELATION    TO    INTERFEROMETRY. 


67 


TABLE   ic  —Values  of  dO/dn  and  displacements  N. 

refer  to  centers  of  ellipses. 


=  .68  cm.     Nc  and  yc,  etc. 


Spectrum  line 

B 

D 

E 

F 

G 

Equation  28.  .. 

deo/dn  = 

i'  27* 

i'     5" 

53". 

46" 

en* 

$ 

27... 

dd/dn  = 

3'    4 

1    52 

i    20 

59 

6° 

dO/dn  = 

—  6'  53* 

-10'  38" 

oo 

+  8<47" 

+  3'  Si" 

'  j 
dfi 

dX 

281 

445 

622 

793 

1140 

d9/dn  = 

—  i'  54" 

—  i'  So" 

QO 

+  2'  21* 

+      40* 

-  i'  44" 

-i'  44" 

oo 

+  2'  n* 

+      36* 

.  . 

35--- 

'.'.'.'.'.'.'.'.'...  ^yc  = 

I.I779 
—  .  0142 

i  .1811 
—  .0110 

i  .  1921 

±  .0 

i  .  1981 
-f  .0060 

i  .2090 
+  .0169 

N  = 

.9235 

.9274 

•9391 

•9456 

•9578 

ANC  — 

-  .0156 

—  .0117 

±  .0 

-f  .0065 

+  .0187 

34*.- 

(observed)  A  Nc  = 
dd/dn  = 

-  -0153 

—  .0098 

-  i'  54" 

±  .0 

oc 

+  .0057 

2'    2l" 

'48* 

Since  equation  (27)  is  clearly  inapplicable,  giving  neither  maxima  nor 
counting  the  fringes,  it  follows  that  in  this  equation  y>en/co&  R;  i.e.,  the 
centers  of  ellipses  are  not  in  correspondence  with  the  path  difference  zero. 
In  other  words  the  air-path  difference  is  larger  than  the  glass-path  difference 


FIG.   29.— Chart  showing  dispersion  per  fringe  in  terms  of  wave-length. 


•"Interpolated  between  D  and  G  by  fi  =  a  +bi  +cA3  where  b=  -  .00273,  c  =  -0000197. 


68  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

in  such  a  way  that  de/dn  is  equal  to  °c  for  centers  (here  at  the  E  line),  but 
falls  off  rapidly  toward  both  sides  of  the  spectrum. 

There  is  another  feature  of  importance  which  must  now  be  accentuated. 
In  case  of  different  colors,  and  stationary  mirrors  and  grating,  y  is  not 
constant  from  color  to  color,  whereas 

2N  =  2y—2e/ji  sin/?  tanR 

is  constant  for  all  colors,  as  has  been  shown  above.  Thus  the  equation  to 
be  differentiated  for  constancy  of  adjustment  but  variable  color  loses  the 
variable  y  and  becomes 

n\  =  2€n  cos  R—  zN  (29) 

N  here  is  the  difference  of  perpendicular  distances  to  the  mirrors,  M  and 
N,  from  the  ends  of  the  normal  in  the  glass  plate,  at  the  point  of  incidence 
of  the  white  ray.  Performing  the  operations 

dl  X2  cos  R 


dn~  2(ycosR-efi(i  +a)) 
d0          tf  cosR 


dn     2DcosO     e{i(i  +a)  —  y  cos  R 
Hence  maximum  dd/dn  =  <*>  at  the  centers  of  ellipses  which  occur  for 

y=    c'os^  (32) 

The  value  of  dd/dn  for  the  different  Fraunhofer  lines  above,  if  a  =  .015  is 
still  considered  constant  and  e=  .68,  is  given  in  table  15. 

These  results  show  that  the  distance  apart  of  fringes  on  the  two  sides  of 
the  center  of  ellipses  is  not  very  different,  though  they  are  somewhat  closer 
together  in  the  blue  than  in  the  red  end  of  the  spectrum,  as  observed. 
There  is  thus  an  approximate  symmetry  of  ovals,  and  dd/dn  falls  off  very 
fast  on  both  sides  of  the  infinite  value  at  the  center.  This  is  shown  in  fig.  29. 

The  observed  angle  between  the  Fraunhofer  lines  D  and  E  for  the  given 
grating  was  4,380".  The  number  of  fringes  between  D  and  E  would  thus 
be  even  less  than  4,38o"/638"  =  6.7  only,  which  is  itself  about  4  times  too 
small.  The  cause  of  this  is  then  finally  to  be  ascribed  to  the  assumed  con- 
stancy of—a  =  (dn/n)/(d\/\),  and  a  discrepancy  is  still  to  be  remedied. 
We  may  note  that  a  does  not  now  enter  as  directly  as  appeared  in 
equation  (27). 

By  replacing  a  by  its  equivalent,  equation  (31)  takes  the  form 

dd          X2  i 

dn=2Dcos6     I*I7~~^yl""  (33) 


and  a  definite  series  of  values  may  be  obtained  by  computing  dn/d\;  but 
as  all  experimental  reference  here  is,  practically,  not  to  path  differences 


IN  RELATION  TO  INTERFEROMETRY.  69 

but  to  displacements  of  the  movable  opaque  mirror  N,  the  form  of  the 
equation  applicable  is 

dd  _  _  P  i 

dn~2Dcosd       '  i       ^-  ~  (34) 


(A       du\ 
ucosR .5   JT)— A7 
cos  R  dX  I 


e 

To  make  the  final  reduction,  I  have  supposed  that  for  the  present  pur- 
poses a  quadratic  interpolation  of  n,  between  the  B  and  the  G  lines  of  the 
spectrum,  would  suffice.  Taking  the  E  line  as  fiducial,  I  have  therefore 
assumed  an  equation  for  short  ranges,  corresponding  to  Cauchy's  in  simpli- 
fied form 

in  preference  to  the  more  complicated  dispersion  equations.  From  the 
above  data  for  light  crown  glass  we  may  then  put,  roughly, 

6  =.456X10-*°  dfjL/dl=-2b/l5 

Thus  I  found  the  remaining  data  of  table  15.  The  results  for  dQ/dn  agree 
as  well  with  observations  as  may  be  expected.  The  ovals  resemble  ellipses, 
but  are  somewhat  coarser  on  the  red  side,  as  is  the  case. 

TABLE   16.— Values  of  dX/dy,  dd/dy,  etc.    e  =  .68  cm. 


B 

D 

E 

F 

G 

Equation 

36 
38 

d^/dy 
dXJdN 

._ 

.00372 
269 

.00318 
229 

.00283 
205 

.00260 
189 

.00229 
167 

cm. 
cm. 

39 

dX/dN 

= 

-  .017 

-  .031 

QO 

+  .030 

+  .015 

cm. 

42* 

dX/dN 

^r 

-  -0055 

OO 

+  .0081 

+  •  o°3  i 

cm. 

42 

dX/dN 

ma 

-  .0044 

—  .0050 

00 

+  .0075 

+  .0023 

cm. 

42 

dd/dN 

= 

-14-6 

-  17.0 

QO 

+  26.0 

+  8.1    radians 

The  centers  of  ellipses  are  thus  defined  by  the  semi-air-path  equation 


?  cos/  '  (35) 

or  the  corresponding  equation  in  terms  of  Nc.  The  trend  data  for  JNC 
agree  fairly  well  with  observation,  except  at  the  D  line,  which  difference  is 
very  probably  referable  to  the  properties  of  the  glass,  since  the  grating  was 
not  cut  on  light  crown. 

The  number  of  fringes  between  the  D  and  E  lines  now  comes  out  plaus- 
ibly, being  less  than  4,  380"  '/  104"  =  42.  It  is  difficult  to  count  these  fringes 
without  special  methods  of  experiment;  but  the  number  computed  is  a 
reasonable  order  of  values,  about  25  to  30  lines  being  observed. 

Some  estimate  may  finally  be  attempted  as  to  the  mean  displacement 
of  mirror  5N  per  fringe,  between  the  D  and  E  lines.  As  their  deviation 
is  6=  i°  13'  and  the  displacement  from  D  to  E,  JNe, 

dd   JNC          )?  i     JA/*C 


_  __  _ 

0/(dd/dn)  ~dn      6       2D  cos~0  JWC     0 


*  Constants  interpolated  between  D  and  G  by  n  =  a+bl  +  cP  when  6=  -.00273. 
c  =  . 0000197. 


70 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


if  the  value  of  dB/dn  for  the  D  line  be  taken.    Thus 

5N>\*/(2D8  cos  0)  =  X2/£>  sin  26,  nearly 

Hence  8N  is  independent  of  the  thickness,  e,  of  the  plate  of  the  grating,  as 
I  showed*  by  using  a  variety  of  different  thicknesses  of  compensator. 
Since  X  =  .000059  cm.,  D  =  . 0003 51  cm. 

dN  >. 0003 1  cm. 

The  values  found  were  between  .00033  and  .00039,  naturally  difficult  to 
measure,  but  of  the  order  required. 


FIG. 


f    •*  I "     w  ' ^   '~ '  -      >«r  I 

30. — Chart  showing  dispersion  (dl/dN),  per  cm.  of  displacement  of 
mirror  in  terms  of  wave-length. 

*  American  Journal  of  Science,  xxx,  1910,  p.  170. 


IN    RELATION    TO    INTERFEROMETRY.  71 

45.  Case  ofd\/dy,  and  dO/dy,  etc. — If  in  equation  (23)  e  and  ware  constant 
while  n,  R,  y,  and  X  vary,  the  micrometer  equivalent  of  the  displacement  of 
fringes  may  be  found.  Here 

which  coefficients  are  given  by  equations  (24),  (25),  and  (26).     In  this  way 

^=  ^cos^  (**•) 

dy        efjL(i+a(i-tan2R))-ycosR 

and 

de/dy  =  -  (d\/dy)/D  cos  6 

If  y  =  efi/cos  R,  where  y  is  variable,  the  motion  is  supposedly  referred  to  the 
centers  of  ellipses.  Thus 

ddn  X  cos  R 

"_ (77) 

dy      D  cos  6  aefi(i  —  tan2  R} 

Values  are  given  in  table  4  and  hold  for  the  glass  of  tables  11-15.  These 
results  are,  as  usual,  many  times  too  large  and  they  contain  no  suggestion 
of  opposed  motion  on  the  two  sides  of  the  center. 

If  we  consider  the  displacement  of  the  mirror  N  instead  of  the  path  differ- 
ence and  for  a  given  color  write 

N=y  —  e  sin  I  tan  R 
then 

dN  efi(i  +a)  —  y  cos  R 

dy  ~etu(i  +  a(i— tan2  jR))  — j>  cos /? 
and  hence  in 

dX0         X  cos  R 

ri\f= (3°) 

the  factor  (i  —  tan2  R)  is  removed  from  the  equation.  But  the  data  are 
not  much  improved. 

The  equation  n\  =  2e/j.  cos  R  —  N  gives,  on  differentiation  and  reduction, 

dX  _  X 

'dN~y-ett(i+a)/aa*_R 

To  be  consistent  with  the  preceding  paragraphs,  it  is  therefore  necessary 
to  put  y— £/*(i+a)/cos  R  =  o,  for  centers  of  ellipses,  so  that  (d\/dN)0  =  (X>  • 
If  a  is  constant  this  supposition  leads  to  results  which  are  throughout  out 
of  the  question.  The  values  of  d\/dN  may,  however,  be  found  by  inserting 
the  data  for  dn/d\  given  in  the  preceding  paragraph. 
Centers  thus  correspond  to 

eX      d,n 


cs  (40) 

or 

dX  =  * 

d N     \ - efi  cos~R  +  e\ (dftjdK) /cos  R 


72  THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 

so  that  if  d\/dN=  «  ,  the  maximum  at  the  centers  of  ellipses,  the  simulta- 
neous effect  at  X'  will  be  (as  the  mirror  has  not  moved) 


D       ,        DA       /     x      dft        /'       dn'\ 

e(ttcosR  —  ft  cos  R')  —  e[         n    .,—        ...  -...-) 

Vcos  R  dX     cos  /^x  d/T  / 


If  the  center  of  ellipses  is  at  the  E  line  the  values  of  table  4  hold.    The 
motion  on  the  blue  side  of  the  E  line  is  thus  larger  than  the  simultaneous 
motion  on  the  yellow  side,  conformably  with  observation. 
All  the  results  are  given  graphically  in  fig.  30. 

46.  Interferometry  in  terms  of  radial  motion.  —  Either  by  direct  obser- 
vation or  combining  the  equations  (34)  and  (41)  for  dX/dn  and  d\/dN  the 
usual  equation  for  radial  motion  again  results 


dn      2 

where  N  is  the  displacement  of  mirror  per  fringe.  This  equation  is  best 
tested  on  an  ordinary  spectrometer,  by  aid  of  a  thin  compensator  of  micro- 
scope glass  revolvable  about  its  axis  and  placed  parallel  to  the  mirror  M. 
The  change  of  virtual  thickness  e'  for  a  given  small  angle  of  incidence  i  may 
then  be  written. 

, 


nearly.    If  /  =  o,  dP  =  (dl)z.    Therefore  2de'=  dP/^.     In  a  rough  trial  for 
e'=.  0226  cm.,  dl  =  .  053  radians,  ^=1.53,  one  fringe  reappeared.     Hence 


/i^'  =  ^X  1.53  X27  X  io~6=  21  X  io-6cm. 
which  is  of  the  order  of  half  the  wave-length  used. 

47.  Interferometry  by  displacement.—  In  a  similar  experiment  the  dis- 
placement of  ellipses  due  to  the  insertion  of  the  above  glass,  e'  =  .0226  cm., 
was  from  the  D  line  to  about  the  G  line.  If  AN  is  the  displacement  of  the 
mirror  N,  to  bring  the  center  of  ellipses  back  to  the  same  line,  D  or  E,  we 
may  write  n  =  i  -\-JN/e'.  I  found  at  the  E  line  /*  =  i  .53  ,  at  the  D  line  n  = 
1.53  ;  special  precautions  would  have  to  be  taken  to  further  determine  these 
indices. 

Thus  there  are  two  methods  for  measuring  /u,  either  in  terms  of  the  radial 
motion  of  the  fringes  or,  second,  in  terms  of  the  displacement  of  the  fringes 
as  a  whole.  Moreover,  paragraph  44  may  be  looked  upon  reciprocally, 
as  a  method  for  measuring  dn/d\,  directly. 


IN    RELATION    TO    INTERFEROMETRY. 


73 


PART  IV.    INTERFERENCES  IN  GENERAL,  AND  SUMMARY. 

48.  The  individual  interferences.— In  figs.  31,  32,  33,  gg  is  the  face  of  the 
grating,  M  and  N  the  opaque  mirrors,  and  I  the  incident  ray. 

As  the  result  of  reflection  from  the  top  face,  the  available  air-path  being 
>>m  and  ym' ',  there  must  be  two  images  of  the  slit  seen  in  the  telescope  directly, 
viz,  a  and  c,  fig.  31.  Of  these  c  will  be  more  intense  than  a,  which  is  tinged 
by  the  long  path  in  the  glass.  These  two  rays  together,  on  diffraction,  will 
produce  stationary  interferences  whose  path  corresponds  to  the  equation 

n\  =  2e/j.  cos  R 
The  optical  paths  of  the  two  rays  are 

reflected-refracted,  I',  2ym-\-en(2  cos  R— sec  R) 

refracted-reflected,  II',  2ym'+T>e(j.  sec  R  =  2ym+en(3  sec  R~4  sin  R  tan  R) 

If  the  plate  of  the  grating  were  perfectly  plane  parallel,  the  slit  images 
a  and  c  would  obviously  coincide. 


FIG.  31. — Diagram  of  interferences  reflected  from  same  face 
of  grating  plate.   Non-compensated. 

The  directly  transmitted  rays,  however,  after  reflection  from  N  give  rise  to 
four  images  of  the  slit — in  case  of  a  slightly  wedge-shaped  plate,  the  one  at 

a,  fig.  3 1,  being  white,  that  at  c  yellowish,  the  distances  apart  being  the  same 
as  in  the  preceding  case.    Besides  this  there  are  two  images  of  the  slit  at 

b,  figs.  32  and  33,  which  result  from  double  diffraction  at  the  lower  face,  the 


74 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


case  shown  corresponding  to  8<i  first  with  9>i  thereafter,  while  the  other 
image  corresponds  to  6  >  i  and  6  <i  in  the  two  successive  diffractions.  These 
slits  show  strong  chromatic  aberration  but  are  nevertheless  linear  and 
useful.  The  optical  paths  for  the  three  lower  rays  are 

transmitted,      I.    2yn+en/cos  R 
transmitted-refracted,     II.    2yn+en/cos  R+zep  cos  R 
transmitted-diffracted,  III.    2yn+en/cos  R-\-ien  cos  0t 

These  three  equations  determine  the  three  stationary  interferences 
nX  =  20/i  cos  R        «X  =  2en  cos  Q\        «X  =  ten  (cos  B^  —  cos  R) 


Jlf 


FIG.  32. — Diagram  of  interferences  reflected  from  both  faces 
of  grating  gg.  Excess  of  glass  path  on  N  side.  Self- 
compensated. 

discussed  in  the  preceding  paper*.  If  the  grating  is  between  two  identical 
plates  of  glass,  these  stationary  fringes  may  become  movable ;  for,  inasmuch 
as  there  is  complete  symmetry  between  the  rays  to  be  reflected  from  the 
mirrors  M  and  N,  the  stationary  fringes  become  identical  for  both  plates 
of  glass,  from  either  of  which  they  may  be  reflected,  before  or  after  dif- 
fraction. Hence  the  motion  of  either  opaque  mirror  changes  the  phase  and 
moves  the  fringes,  which  are  now  linear,  vertical,  and  nearly  equidistant. 

*C.  and  M.  Bams. 


IN    RELATION    TO    INTERFEROMETRY. 


75 


They  may  also  be  regarded  as  confocal  ellipses  of  infinite  size,  the  visible 
parts  of  their  peripheries  lying  close  together  in  the  field  of  view.  They 
pass  through  infinity  in  this  field,  when  the  virtual  path  difference  is  zero. 


FIG.  33. — Diagram  of  interferences  reflected  from  both 
faces  of  grating  plate.  Excess  of  glass  path  on  M 
side.  Self-compensated . 

49.  The  combined  interferences.— The  five  equations  given  by  the 
first  group  I'  and  II'  when  combined  with  the  second  group,  I,  II,  III,  lead 
to  six  other  interferences,  all  of  them  of  the  movable  type  and  useful  for 
interferometry.  If  for  simplicity  the  path  difference  is  zero,  they  may  be 
written,  if  y  =  yn  — ym  and  y'= yn— ym'  and  y0  corresponds  to  the  glass-path 
difference,  N  as  usual  referring  to  the  displacement  of  the  opaque  mirror: 

IV,  Reflection  at  bottom,  equation  I-II',  y0'  =  en/c,Qs  R,  N  =  en  cos  R 
V,  Reflection  at  bottom,  equation  II-I',—  y0  =  en/cos  R,N  =  encosR 

VI,  Reflection  at  top,   equation  III-I',  — ^O  =  ^M(COS  0i+sin  R  tan  R), 

N  =  en  cos  0i 

VII,  Reflection  at  top  and  bottom,  eq.  II-II',  y0' =e\i  sin  R  tan  R,  N  =  o 
VIII,  Reflection  at  top  and  bottom,  eq.  III-II',  y0'  =  en(sec  R-cos  00, 

N  =  en(cos  R—cos  0i) 
IX,  Reflection  at  bottom  and  top,  eq.  I-I',—  y0  =  en  sin  R  tan  R,N  =  o 

Hence  the  fringes  of  the  interference  VII  and  IX  are  identical  through- 
out the  spectrum,  when  the  mirror  N  moves.  The  remaining  fringes  are 
elliptic  and  eccentric,  because  reflected  from  two  faces  of  the  thin  wedge. 


76 


THE    PRODUCTION    OF    ELLIPTIC    INTERFERENCES 


Nos.  VII  and  IX  are  parallel  lines  which  pass  symmetrically  from  negative 
to  positive  obliquity,  or  the  reverse,  respectively,  through  horizontality, 
in  opposed  directions. 

The  results  of  these  equations  have  been  computed  for  light  crown  glass 
and  the  Fraunhofer  lines  B,  D,  E,  F,  G  above  and  in  the  following  table  17. 

TABLE   17. — Diffracted-reflected  and    reflected-diffracted    rays.    e  =  .68   cm.    D  = 
.000351  cm.  7  =  45°,  nearly.  Light  crown  glass,  y^y^-y^.  Change  of  faces:  y- N. 


Line  of  Spectrum                 B                 D                  E 

F 

G 

io«A=      68.7              58.93            52.70 

48.61 

43  -08 

«=        1.5118          1.5153          1.5186 
R=       27*53'         27*49'         27*45' 

1-5214 
27°  42' 

1-5267 
27635' 

0,  =            19°  47'              20°   51'              21°  Jl' 

21°  57' 

22°  30' 

(correction)  2A  =          .5088            -5°74            -5°59 

•5050 

-5023 

I 

.   Reflection  from  a  single  face.    yQ=eft  cos(R  —  01)/cos 

R: 

nl  =  o;-      2y0=        2.3030         2.3128         2.3199 

2.3252 

2-3334 

E*=o;—  2AJ0=    —    .0169       -    .0071      ±    .0 

+    -o°53 

+    -0135 

correction  —  2  <5Ar0  =    4    .0028      +    .0014      ±    .0 

—    .0009 

—    .0036 

—  2AAr0=    —    .0197       -    .0085      ±    .0 

4-    .0062 

4    .00171 

II 

.   Reflection  from  one  face.    y0=efi/cos  R: 

+     2y0=        2.3261          2.3301          2.3337 

2-3369 

2.3426 

4-   2A.y0=    —    .0076      -    .0036     ±    .0 

4-    .  003  2 

4-    .0089 

+  2AA0=      —      .OIO4          -      .0050        ±      .0 

4-    .0043 

4-    .0126 

III 

.   Reflection  from  both  faces.     y0  =  en  (sec  R  —  cos  0j)  : 

+      2^0  =          .39J3            -4043            .4124 

•4177 

-4242 

+    2AV0  =      —      .0211        —      .Oo8l         ±      .0 

+    -0053 

4    .0118 

4-2AAr0  =      -      .0239         -      .0095         ±      .0 

4-    .0062 

+    .0154 

IV 

.  Reflection  from  both  faces.    yt  =  ep  sin  0stan  R: 

+    2y0  =        .3682         .3870         .3985 

.4061 

•4151 

4-     2Ay0=      —      .0303          -      .0115                   .0 
+  2AAo=      —      .0332        —      .0130                 .0 

4-    .0075 
4-    .0084 

4    .0166 
4    .0202 

V 

.  Same  as  II,  but  reflection  from  top  face  : 

-     2y0=        2.3261          2.3301          2.3337 

2-3369 

2.3426 

—     2Ay0  =      —      .0076         —      .0036         ±      .0 

4-    .  003  2 

4    .0089 

—  2AA70=      —      .OIO4          -      .OOSO         ±      .0 

4     .0043 

4     .0126 

50.  Special  results.— The  table  shows  the  following  positions  for  the  suc- 
cessive interferences,  i.e.,  for  zy—N  when  the  reflection  changes  from  the 
lower  to  the  upper  face.  N  =  2eu  tan  R  sin  R. 


Reflection  from   bot- 
tom, ym<yn 

Reflection  from    top, 


[Cases     I-IP,  yn  =  1.1650,  Equation     IV,  Position,    1.1650 

II-IP,  .2537,                     VII,                        .2537 

Ill-IP,  .2021,                   VIII,                          2021 

I-  P,  -    .2537,                      IX,                        .3537 

II-  P  - 1 . 1650,                         V                   -      6576 

?-»'»                          I           III-P,  -1.2166,                       VI,                  -    ^092 

Hence  the  total  play  of  the  mirror  N  should  be  1.1650  — (  —  .7092)  =1.8742 
cm.,  with  which  the  observed  datum  1.88  agrees  as  nearly  as  the  experiment 
warrants.  Again  the  fringes  of  equations  VII  and  IX  are  simultaneously 
in  the  field,  which  is  true,  these  being  a  series  of  lines  which  change  their 
obliquity  in  opposite  directions  symmetrically.  The  two  ellipses  of  equa- 
tions V  and  VI,  however,  were  found  nearer  together  than  the  com- 


IN  RELATION  TO  INTERFEROMETRY.  77 

puted  value.  Thus  their  distance  apart  in  the  D  region  corresponded  to 
J;y  =  .oo75  cm.;  in  the  E  region  it  corresponded  to  J^  =  .oo5o  cm.,  whereas 
the  above  data  require  .0516  cm.  Here,  however,  there  is  some  confusion. 
For  instance,  the  spectra  may  coincide,  whereas  the  doubly  diffracted 
white  and  colored  images  of  the  slit  do  not.  Moreover,  there  are  three 
ellipses  in  this  region  of  y,  since  the  double  diffraction  for  6>i  also  gives  an 
ellipse,  not  included,  because  not  belonging  to  the  above  enumeration. 
The  motion  of  ellipses  from  the  D  to  the  E  line  has  different  values.  The 
set  toward  the  violet  corresponds  to  JN  =  .oogo  cm.;  the  set  toward  the 
red  to  JN=  .0070  cm.  There  is  thus  a  different  speed  of  fringes  relative  to 
N,  and  closer  contact  in  the  violet  than  in  the  red.  The  latter  are  also  more 
nearly  circular  than  the  former.  Equation  IV  gives  the  strong  solitary 
interferences  useful  in  practice  and  treated  in  detail  in  the  earlier  parts  of 
this  paper.  The  treatment  of  the  other  cases,  being  less  important,  may 
be  omitted  here. 

My  thanks  are  due  to  Prof.  Joseph  S.  Ames,  of  Johns  Hopkins  University, 
for  his  kindness  in  lending  me  the  glass-diffraction  grating  by  which  the 
above  equations  were  tested.  I  hope  at  some  other  opportunity  to  work 
with  a  grating  whose  refraction  is  known  throughout  the  spectrum  and 
also  to  endeavor  to  obtain  the  phenomenon  as  clearly  from  film  gratings 
(replicas)  as  has  been  possible  for  the  linear  series  in  the  preceding  paper 
(I.e.).  Thus  far  the  above  phenomena  as  obtained  from  film  gratings  were 
not  strong  and  sharp  enough  for  measurements  of  precision  but  in  a  series 
of  experiments  which  has  since  been  completed  I  will  show  how  this  desid- 
eratum may  be  realized. 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 

Los  Angeles 
This  book  is  DUE  on  the  last  date  stamped  below. 


Form  Ly-50m-ll,'50U554)444 


000702610    7 


UNIVERSITY  of  CALIFORNIA 


A  wnwi  . 


